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					IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 3, MAY 1994                                                                 743




                                                  Perfect Maps
                                                       Kenneth G. Paterson


   Abstmct-Given positive integers r , s, U , and U, an ( r ,s; U,U)   question addressed in this paper is, given U and U, for
Perfect Map (PM) is defined to be a periodic r X s binary array        which r and s does there exist an ( r , s; U ,U ) PM? In [SI,
in which every U X U binary array appears exactly once as a
periodic subarray. Perfect Maps are the natural extension of the       Mitchell and this author gave three simple necessary
de Bruijn sequences to two dimensions. In this paper the exis-         conditions for the existence of an ( r , s; U,U ) PM:
tence question for Perfect Maps is settled by giving construc-             Result 1.1 ([18], Lemma 2.1): Suppose there exists an
tions for Perfect Maps for all parameter sets subject to certain       ( r , s; U ,U ) PM, A. Then
simple necessary conditions. Extensive use is made of previously              i) rs = 2"",
known constructions by finding new conditions which guarantee
their repeated application. These conditions are expressed as                ii) r > u o r r = u = l ,
bounds on the linear complexities of the periodic sequences                iii) s > U or s = U = 1.
formed from the rows and columns of Perfect Maps.                      In view of Result 1.1 we will often write r = 2 k and
                                                                          = 2uu-k , where 0 s k I: uu and Z k > U, 2uu-k > U , so
Index Tenns-Perfect Maps, de Bruijn array, window property,
linear complexity.                                                     that we will refer to ( Z k , 2U"-k;U ,U ) PMs.
                                                                           The gap between these necessary conditions and the
                                                                       previously known constructions is wide. Reed and Stewart
                        I. INTRODUCTION                                [2] gave an example of a (4,4; 2 , 2 ) PM, and showed that in
                                                                       this simplest of cases there are essentially only two dif-
G      IVEN positive integers r, s, U , and U , an ( r , s; U ,U )
       Perfect Map (PM) is defined to be a periodic r X s
binary array in which every U X U binary array appears
                                                                       ferent PMs. Ma [lo] and Fan et al. [ll], using the termi-
                                                                       nology "de Bruijn arrays," noted that a PM with either r
                                                                       or s equal to 1 is simply a de Bruijn sequence of span U or
exactly once as a periodic subarray.
                                                                       U, respectively. They showed that given U and U , there
   These arrays have applications in diverse areas of com-
                                                                       exist r and s for which an ( r , s ; u , u ) PM can be con-
munications and coding, [ll, such as two-dimensional range
                                                                       structed. They also gave two recursive constructions under
finding, scrambling, and various kinds of mask configura-
                                                                       which ( r , s; U ,U ) PMs satisfying certain constraints may be
tions. Perhaps the most obvious use for such arrays is in
two-dimensional range finding (or position location). The
                                                                                                  +
                                                                       used to obtain ( r ,2"s; U 1,U ) PMs and (Zr, 2"-'s; U +
                                                                        1, U ) PMs. They justified repeated use of these construc-
basic idea is that, if an ( r , s; U ,U ) PM is written in some
                                                                       tions using ingenious special arguments and showed that
way onto a planar surface of dimensions r by s, then any
                                                                       there exists a ( 2 u 2 / 2 , 2 u ' U ,U) PM whenever U is even.
                                                                                                      1;
device capable of examining a U by U rectangular subar-
                                                                       In [l] Etzion gave a construction for a large class of
ray will be able to determine precisely its position on the
                                                                        Perfect Maps, settling the existence question for
surface. Brief mention is made of such an application in               (2k, p - k . ,U ,U ) PMs whenever U < Z k I 2" except when
[2] and a more detailed description of applications of this
                                                                       U = k and U = 2. His constructions depend on the exis-
type can be found in [3].Note also that, in the one-dimen-
                                                                        tence of Hamiltonian cycles in certain graphs; Etzion gave
sional case, similar position-detection applications have
                                                                        an algorithm for finding such cycles, while [8] gave an
been suggested for de Bruijn and m-sequences by a
                                                                        explicit construction for such cycles in terms of 2k-ary de
number of authors (see for example, Bondy and Murty [4],
                                                                        Bruijn and related sequences.
Petriu and co-workers [5], [61, and Arazi [71). To be useful
                                                                            In this paper, we show that the conditions of Result 1.1
in this type of application, the decoding problem, i.e.,
                                                                        are in fact also sufficient for the existence of Perfect
resolving the position of a particular subarray within a
                                                                        Maps. In other words, we settle the existence question for
specified Perfect Map, is of great significance. This topic
                                                                        Perfect Maps. We make extensive use of the constructions
is addressed in [8].
                                                                        of Fan et al., but we give them more power by examining
   Perfect Maps are also mathematically interesting in
                                                                        conditions which guarantee their repeated applicability.
their own right as the natural extension of the binary de
                                                                        These conditions are expressed as upper bounds on the
Bruijn sequences [9] to two dimensions. The fundamental
                                                                        linear complexities of the rows and columns of Perfect
                                                                        Maps. We outline some basic notation and give the neces-
  Manuscript received August 14, 1992; revised March 1, 1993. This      sary background on linear complexity in Section 11, while
work was supported by Hewlett Packard Ltd. through SERC CASE
award no. 90(3/11574 held at Royal Holloway, University of London,      in Section I11 we give an analysis of the recursive con-
and was presented at the 1993 IEEE International Symposium on           structions in [ll], and an indication of how they might be
Information Theory, San Antonio, TX,Jan. 17-22, 1993.                   used to construct large classes of Perfect Maps starting
  The author is with the Department of Mathematics, Royal Holloway,
University of London, Egham, Surrey TW20 OEX,U.K.                       from a basic class. In Section IV, we review the work of
  IEEE Log Number 9401369.                                              Etzion in [l]; in particular, we study the linear complexi-

                                                    0018-9448/94$04.00 0 1994 IEEE
744                                                       IEEE TRANSACTIONSON INFORMATION THEORY, VOL. 40, NO. 3, MAY 1994


ties of the sequences of the Perfect Factors which form            Result 2.2 ([12]): Suppose (s) has least period 2k. Then
the building blocks of Etzion's construction.
                                                                                       2k-' + 1 I c(s) I 2k
   In Section V we give explicit examples of some special
PMs with low U and U, which are of great use in settling         and c(s)   =   2k if and only if (s) has odd weight.
the existence question for PMs with 2 I U , U I 5 by             Define the operator D by D = E + 1. Then for (s)                                     E
means of a case by case discussion. The next step in our
                                                                 S(p),
argument is to construct a suitable basic class of PMs to
which the construction of [ll]may be repeatedly applied;           D ( s ) = (so + s1, s1     + S2,"',      sp-2    + s p - l , s p - l + so>.
these constructions comprise Section VI. In the penulti-         In fact D is simply Lempel's homomorphism acting on
mate section, we complete our programme by using the             periodic sequences-see [13] for details. We use the fol-
basic class of PMs to produce PMs for all parameter sets         lowing result of Lempel:
satisfymg the necessary conditions. Finally we discuss some        Result 2.3 ([13]): Suppose (s) E S ( p ) has even weight.
related open problems in this area.                              Then the set of preimages of (s) under D-17 denoted by
                     NOTATION RESULTS
             11. BASIC      AND
                                                                 D - ' ( s ) , consists of a pair of complementary sequences of
                                                                 period p given by
   We follow the notation of [121, the results of which are
                                                                                                            P-2
fundamental to this paper. Let S ( p ) denote the set of
                                                                 D-'(s)=       ((o,so,so+s ',**.,
binary sequences satisfying si = si+,, for every i 2 0. We
say that the sequences of S ( p ) have period p . If p is the
                                                                                                            zs1/,
                                                                                                            r=O

least positive integer for which sequences (s) satisfies
si = s i + p ,
             then we say that (s) has least period p . We may
represent a member (s) of S ( p ) by its generating cycle:
                                                                                         i1,1   + so,1 + so + S1,"', 1 +
                                                                 When (s) has odd weight,'then D - ' ( s ) contains one
                                                                                                                                         p =2
                                                                                                                                         I-o
                                                                                                                                                s,
                                                                                                                                                     11.


                  (s) = (so,       S1,'",    sp-J                sequence of period 2 p given by
Suppose that for all i 2 0, (s) satisfies the linear recur-
rence
                                                                 D - (s)   =    (   (0, so,so + s1   ;e*,
                                                                                                            P-2

                                                                                                            r=O
                                                                                                                  s, ,1 , 1   + so, 1 + so
                             m
                        +
                 Si+"
                            j= 1
                                   ajs[+"-j      =   0,

of degree m over the field GF(2). Then if E denotes the
                                                                                                                    +s,;**,      1   +
                                                                                                                                         r=O
                                                                                                                                                s,
                                                                                                                                                     i).
                                                                 It is clear from the definition of D and Result 2.1 above
shift operutor whose action on (s) is defined by                 that when p is a power of 2, an application of D to a
                                                                 sequence (s) E S ( p ) produces a sequence having linear
                            Es,    =si+l,
                                                                 complexity one less than that of (s), and conversely the
we have, for all i 2 0,                                          sequence(s) in D - ' ( s ) have linear complexity one greater
                                                                 than that of (s).
                             m
                                                                                       OF FAN,
                                                                     111. THECONSTRUCTIONS   FAN,MA,AND SIU
                            j= 1
                                                                     In this section we examine the recursive constructions
i.e.,                                                            for Perfect Maps given in [ l l ] and show how linear
                          f(E)Si      =     0,                   complexity of the sequences formed from rows and
                                                                 columns of Perfect Maps can be used as a tool to control
for a polynomial f of degree m . f ( E ) is called the           the repeated application of these constructions.
minimalpolynomial of (s) when f ( E ) s , = 0 and f has the          Construction 3.1 ([ll], Construction 5.1): Let A be an
least possible degree. In this case, the linear complexity of    (r, s; U , U ) PM whose column weights are even (i.e., re-
sequence (s) is defined to be the degree of the minimal          garding the columns of A as s sequences of period r,
polynomial of (s), and is denoted by 4s). It can be shown        each sequence has even weight). Then there exists a
that the minimal polynomial divides any other polynomial         ( r ,2"s;U + 1, U ) PM B.
satisfying g ( E ) s , = 0. Clearly when (s) E S ( p ) , (EP +       The construction proceeds by considering the columns
l)s, = 0 and so f(E)IEP + 1.                                     of A as s binary sequences of period r and even weight,
   Suppose that p = 2k. Then E2' + 1 = (E + 1)2kbe-              applying the operator D-' to them to obtain a set of 2s
cause the field characteristic is 2, and so f ( E ) = (E +       sequences of period r and then using this new set as the
Wr). implies:
       This                                                      columns of B in such a way as to ensure that every
  Result 2.1 ([12/): With notation as above, c(s) = c if         ( U + 1) X U matrix appears once in B.
and only if (E + 1)'-'si = 1, for all i 2 0.                         The ordering of columns is specified as follows. Let
We define the weight of a sequence (s) E S ( p ) to be the                                         )
                                                                 ( d ) = ( d o , . . . , d 2 L - lbe a span U binary de Bruijn se-
sum so + s1 +          + s ~ - ~ Note that the period p here
                                 .                               quence commencing with U zeros so that (dY =
need not be the least possible period of (SI.                                ,
                                                                 ( d l ; * . d 2 L - is the punctured de Bruijn sequence derived
                                                                                   1)
PATERSON: PERFECT M A P S                                                                                                                                         745


from ( d ) by deleting one of the initial run of zeros. Define                even weight except in the case                     U =   1 where (d)' = (11,
a sequence ( b ) to have generating cycle                                     and so we have
       ( b ) = (O,.**,O, d l , * * d 2 " - 1 , * . * , l , * *d2L1)
                                   *,              d          .,                 b,   + bi+s+ bi+2s+           * e *   +bi+(2L-1)S 0
                                                                                                                                 =            (U          2 2).

consisting of a run of s zeros followed by s copies of the                    Hence when U 1 2, (b) satisfies a linear recurrence of
sequence (dY, so that ( b ) has least period s (2" - 1)s  +                   degree (2" - 1)s and SO
= 2"s.
                                                                                               c ( b ) 5 (2" - 1)s = 2u2-k              - 2uu-k
  Then the array B is obtained by taking as its Zth
                                                                                                                      -~
                                                                                                       - 2 ( ~ + 1 ) ~- k U V - k
column the preimage under D-' of the Zth column of the
array [AI AI    IA] (the array consisting of 2" copies of A
              0 . .
                                                                              When U = 1, (b) = (O;.-,O, l;.., 11, a sequence consisting
placed side by side), beginning with bit b,. Thus, the first                  of two runs each of length 2 u - k - 1 and having linear
row of B is the sequence ( b ) and subsequent rows are                        complexity 2 u - k I.    +
obtained as the cumulative sum of ( b ) with the rows of                         We now examine the other rows of B. They are ob-
[AIAI... \ A ] .                                                              tained as the cumulative sum of (b) with the rows of
   A proof that Construction 3.1 yields a PM B with the                       [AIAI    0 . .   IA]; i.e., if B has j t h row (b)' =
desired parameters is to be found in [111.                                    (bd,b{,.,.,b$LS-l)          and [AIAI-.. IA] has jth row (a)' =
   Note that by Result 2.2, the column weights of A are                       (ab, a { ; . - ,a $ - l ) , then
even if and only if the linear complexities of the columns
of A are nonmaximal, i.e., strictly less than the upper                     b,' = b, + a: +        +ai-'          ( j 2 1).
bound of Result 2.2. Also, applying D-' to the columns For j 2 1 and any i,
of A results in each column of B having complexity one
greater than the corresponding column of A. Thus the                                                             +
                                                                    (E + l)c(b)-'b,' = (E + l)c(b)-'(bz U : + +ai-')                          * * a



possibility of making, say, t repeated applications of Con-
struction 3.1 depends on the original PM having all col-
umn complexities at least t less than the period. Clearly                               .(a; +          +ai-').          * a *


the determination of linear complexities is an important
step in the construction of Perfect Maps. In what follows, Now ( b ) has linear complexity c(b) and least period
                                                                  2~2uu-k so by Result 2.2,
                                                                          ,
we obtain bounds on the complexities of the rows and
columns of the array B resulting from Construction 3.1.               c(b) - 1 2 2u-12~U-k > 2uu-k
                                                                                                 -                 (since U 2 1).
We begin with a lemma whose proof follows from the
above discussion.                                                 But the sequences ( a ) k have least period 2u"-k or less
                                                                                                +
   Lemma 3.2: Let A be a ( 2 k , 2 u u - k ; ~PM ) whose and so (E l)c(b)-laf = 0 for all i 2 0 and k 2 1. Then
                                                     ,~
columns all have linear complexities less than or equal to
                                                                      (E + l)c(b)-'b/ = (E + l)C(b)-lbi (0 + +0)  +
c. Then if B is the PM obtained from A by using
                                                                                                                                                  - * a




Construction 3.1, the columns of B all have linear com-                                = ( E + l)c(b)-'bi
plexities less than or equal to c + 1.
                                                                                       = 1,
   Lemma 3.3: Let A be a ( 2 k , 2 u u - kU;,U ) PM with even
weight columns. Then if B is the ( 2 k , 2 ( U + 1 ) UU- k 1, U )
                                                     ;           +and so by Result 2.1, c((b)') = c(b) for all j 2 1. The
PM obtained from A by using Construction 3.1, the rows lemma follows.                                                               0
of B all have equal linear complexities c satisfying                 We note that if a PM A has even weight rows then
                                                                  Construction 3.1 may be applied to the rows of A to
                      5 2 ( ~ + 1 ) ~-k U V - k
                                     - ~
                                                                  obtain a new PM whose row complexities are increased by
(i.e., the linear complexities of the rows of B are at least 1 over those of A and whose column complexities are
~ U V - kbelow the maximum possible complexity), unless           bounded by an expression similar to that in Lemma 3.3. In
U = 1 when the rows of B all have linear complexities addition, if A is a ( 2 k ,2 U " - kU ,U ) PM with even column
                                                                                                      ;
equal to 2 u - k + 1.                                             weights and B is the ( 2 k , 2 ( u + ' ) u - k + 1, U ) PM obtained
                                                                                                             ;U
      Proofi As observed above, the first row of B is the         from A by using Construction 3.1, then from Lemma 3.3,
sequence ( b ) in Construction 3.1 which in turn is derived the rows of B all have equal linear complexities c satisfy-
from the sequence (dY. Consider for any i 2 0, the ex- ing
pression                                                                              I2(u+l)u-k - ~ U U - k

             b,       + bi+S+ bi+2s + ... +bi+(2c-l)st                (3.1)   By Lemma 3.2 and Result 2.2, we may make between one
                                                                              and 2 u u - k applications of Construction 3.1 to the rows of
where s = 2u"-k.                                                              B to obtain new Perfect Maps, each having column com-
   Since gcd(s, 2" - 1) = 1 and there are s copies of (dY                     plexities that are nonmaximal and bounded by expressions
in a period of ( b )we see that in the sum of (3.1) we pick                   similar to those of Lemma 3.3.
up every term of (dY exactly once each, together with one                       We next discuss a similar result to Construction 3.1
zero from the run of s zeros. But the sequence (dY has                        appearing in [ll]:
746                                                             IEEE TRANSACTIONS ON INFORMATION THEORY,VOL. 40,NO. 3, MAY 1994

    Construction 3.4 ([llJ,         Construction 5.5): Let A be a         all have linear complexity equal to that of (b), and ii)
( r , s ; u , u ) PM having odd column sums (i.e., columns                follows.
having maximal linear complexity). Then there exists a                       For u 3, we suggest the following procedure to define
                 +
(2r, 2"-'s; U 1, U ) PM, B.                                               a suitable sequence ( b )which is equivalent to that of [ll].
    Similarly to Construction 3.1, Construction 3.4 proceeds              Let ( d ) be a spand U - 1 binary de Bruijn sequence
by considering the columns of A as s binary sequences of                  beginning with U - 1 zeros, and let ( e ) be the preimage
period r and maximal complexity, applying the operator                    under D-' of ( d ) which begins with the all-zero u-tuple.
D-' to them to obtain a set of s sequences of period 2r                   Let (eY be the sequence obtained from ( e ) by deleting
(by Result 2.2) and then using this new set as the columns                one of the zeros from the u-tuple and define sequence ( b )
of B in such a way as to ensure that every (U 1) x u       +              to be the concatenation of s zeros and s copies of (e)'.
matrix appears once in B. The ordering of columns is                         When u 2 5, the result of [14] guarantees the existence
specified as follows. Let ( b )be a sequence of period 2'-ls              of a spand u - 1 binary de Bruijn sequence ( d ) having
such that for each i (1 Ii I s, the 2'-'         )             u-tuples                      +
                                                                          complexity 2"-2 u - 1 I2"-' - 2. Then d e ) = c(d)
bj+rs, j + l + r s , . . ~ , b i + u -(lt+ l s0, 1;.*,2"-' - 1) are all
          b                               =                               +   1 s 2"-' - 1 so that ( e ) and hence (e)' have even
distinct and no two are complements of each other. Suit-                  weight. An argument identical to that of Lemma 3.3
able sequences ( b ) are specified in [ l l ] and in the proof                                                    .
                                                                          shows that c(b) I 2"-12UU-k 2uu-k When u = 3 or
                                                                                                           -
below.                                                                    u = 4, the procedure above results in sequences (e)' with
    Then the array B is obtained by taking as its Zth                     odd weight. Similar reasoning to that of Lemma 3.3 shows
column either the preimage under D-' of the Zth column                    that in these cases, for every i 2 0,
of the array [AIAI... IA] (2'-' copies), or its comple-
                                                                                                    +
                                                                                    b, + b,+, + b,+2s   * * e   +br+(2L-i-1)S   =   1.
ment, depending on whether bit b, is 0 or 1. Thus the first
row of B is the sequence ( 6 ) and subsequent rows are the                Then
corresponding cumulative sums.
    We study the linear complexities of the rows and                      4   + bl+l + 4 , s + b,+,+l + 4 + 2 s + br+2s+l +         .**


columns of a PM B resulting from Construction 3.4. Our                                     + br+(2'-1-l)s b,+(2L-I-1)S+I1 + 1 = 0,
                                                                                                       +              =
result, analogous to Lemma 3.2 and Lemma 3.3, is
    Lemma 3.5: Let A be a (2k,2UU-k;,U ) PM having odd
                                                    U
                                                                          so that c(b) I2 U - 1 2 U U - k 2 u u - k+ 1. A similar argu-
                                                                                                       -
                                                                          ment to that of Lemma 3.3 now shows that for each u the
column sums. Then there exist suitable sequences ( b )
such that when Construction 3.4 is used to produce a                      rows of B all have linear complexity equal to that of the
(2k+l,2"- 2U"-k; + 1, U ) PM B from A in conjunction
                        U                                                 appropriate (b).iii) and iv) follow.                       0

with sequence (b),the linear complexities of the columns                                   IV. ETZION'SCONSTRUCTION
of B are equal to 2k + 1, the minimum possible, and if
                                                                             In [l], Etzion gave a construction for class of Perfect
      i)     1, then the linear complexity of each row of B
           U =                                                            Maps with certain parameters. He introduces the idea of
        is at most the greatest of the linear complexities of             a Perfect Factor, a special collection of sequences forming
        the rows of A.                                                    the basis of his construction for Perfect Maps.
    ii) U = 2, then the row complexities of B are 2""-' +                    Result 4.1 ([l], Theorem 4): Let U , k be integers with
        2.                                                                k Iu < 2 k . Then there exists a collection of 2u-k binary
   iii) u = 3 or U = 4, then the row complexities of B are                sequences, each of period 2 k ,with the property that every
        less than or equal to 2c-12uu-k- 2Uc-k+ 1.                        binary u-tuple occurs in a unique sequence of the collec-
                                                                          tion.
   iv) u 2 5, then the row complexities of B are less than
                                                                             We denote such a collection as a (U,k ) Perfect Factor.
                                -
        or equal to 2U-12UU-k 2uu-k.
                                                                          In view of the results of Section 111, we study the linear
      Proofi The linear complexities of the columns of B
                                                                          complexities of the sequences of Etzion's Perfect Factors.
are one greater than those of A, as in Lemma 3.2, and A
                                                                          Because of our need for sequences with nonmaximal
has column complexities equal to 2k. It follows that B has
                                                                          linear complexities, we in fact adapt Etzion's construction
column complexities equal to 2k + I.
                                                                          to obtain the following:
   We now examine the complexities of the rows of B.
                                                                             Lemma 4.2: Let U , k be integers with k IU < 2 k .
When U = 1, a suitable sequence is ( b ) = (0). Then the
                                                                          Then there exists a (U,k ) Perfect Factor whose sequences
first row of B is (0) and the other rows of B are formed
                                                                          have linear complexities
from the cumulative sum of the rows of A. It is easy to
see that the linear complexity of each row of B is less                                +
                                                                              i) 2 k - ' U if k I U < 2 k p ' ,
than or equal to the greatest of the linear complexities of                  ii) U + 1 if zk-' IU < 2k.
the rows of A, and (i) follows.                                                 Prooc The Perfect Factors of case i) are derived in
   When u = 2, a suitable sequence is ( b ) =                             the same way as those of [l], by repeatedly applying D-'
(O,O;..,O,O,I,O, 1,-.-,0,1) consisting of s = 2ur'-k zeros                to a de Bruijn sequence of minimal complexity. For case
followed by s/2 repetitions of the sequence (01). It is easy              ii) we also use the method of [ll, where it is shown that
to see that ( b ) has linear complexity s + 2. A similar                  the set of binary sequences of period 2k and linear com-
argument to that of Lemma 3.3 shows that the rows of B                    plexity U + 1 has the property that every binary u-tuple
PATERSON PERFECT M A P S                                                                                                 747


occurs exactly once as a subsequence of one sequence of                                                              '
                                                                (so that the column weights are even unless U = 2k- or
                                                 ~ )
the set. That is, this set constitutes a ( ~ , 2 Perfect        u = 2 k - 1).
Factor. We note that this class actually exists for the              Proof Because of the bounds on U , U , and k imposed
range 2k-' s U < 2k, whereas in the original proof of           in the statement of the theorem, Result 4.3 guarantees
Result 4.1, the case where U = 2k-' was covered under i).       the existence of a ( 2 k ,2(U-')"-k U - 1,v) PM A . Since
                                                                                                   ;
                                                          0     the sequences of a (U - 1, k) Perfect Factor form the
   From i) and ii) above it is clear that the sequences in a    columns of the PM A it is clear from Lemma 4.2 that the
Perfect Factor can be chosen to have even weight except         columns of A can be chosen to have even weight. Hence
when U = 2k - 1. It is easily shown that a Perfect Factor       we may apply Construction 3.1 to the columns of A to
with even weight sequences cannot exist in this excep-          obtain a (2k,2Uu-k; ,U ) PM B having, in view of Lemma
                                                                                    U
tional case.                                                    3.2 and Lemma 3.3, row and column linear complexities as
   We now give a brief review of the construction of [l], its   given in the statement of the theorem.                 0
relation to Perfect Factors, and Mitchell and Paterson's
[8] version of Etzion's construction.                                             MAPS
                                                                         V. PERFECT   FOR SMALLU         AND U
   Result 4.3 ([I]): Suppose U ,U 2 2. Then there exist         Here we will establish the existence of (2k,2"U-k;U ,U )
(2k,2""-k; U ,U ) PM's for every k satisfying                 PMs when U , U I 5 for k subject to the conditions of
                      U + 1 s 2k s 2 " ,                      Result 1.1 by means of a case by case discussion. This
                                                              discussion is necessary primarily because it is difficult to
except when U = 2, where we require
                                                              extend the constructions of Section VI for our basic PMs
                    U +   1 I2k I 2 " - ' .                   below U = 6. Because of the repetitive nature of our
   The basic idea behind this construction is to consider arguments, we will not present every case here; the inter-
the graph whose vertices are the 2k X U binary matrices ested reader is referred to the Appendix for full details.
of the form (s, E ' ~ ( S ) ~ ,E'c-l(s)"), where (s)i is a
               (),               ...,                           Throughout this section we make use of the fact that by
sequence in a (U,k) Perfect Factor, written vertically, and transposing our arrays we may assume, without loss of
ti is an integer between 0 and 2k - 1. Since there are generality, that U 2 U . It is sufficient then to construct
2"-k distinct sequences in a (U,k) Perfect Factor, there arrays for every U 2 U and k satisfying the conditions of
are 2uu-kdistinct vertices in the graph. Two such vertices Result 1.1. Moreover, when U = U we need only construct
C and D are joined in the graph if the columns of D can PMs for k I u2/2. In fact, in this section we shall con-
all be shifted by the same amount in such a way that its struct PMs for 2 s U I U I 5; PMs for U = 1 will be
first U - 1 columns agree with the last U - 1 columns of constructed in Section VI.
C . Etzion showed that these matrices can be linked up to        We begin by presenting four examples of Perfect Maps
form the desired PM by showing that a Hamiltonian cycle with low U and U whose row complexities and column
exists in the graph above.                                    complexities have special values. The properties of these
   The contribution of [8] was to give an explicit descrip- special PM's will be used to construct certain other PMs
tion of how to choose the relative shifts and the sequences in this section and in Section VII.
of the Perfect Factor. This method has the advantage that       Lemma 5.1. There exists (4,16; 2,3) PM A, with every
an array so produced can be decoded more easily than the column sum and row sum even. There exists (4,16; 2,3)
original Etzion array. That is, given a particular U X U PM A, with every column sum even and every row sum
matrix, it is possible to find the position of that matrix in odd.
the r X s PM. The construction is a generalization of              Proof Let (s) be the spand 2 binary de Bruijn se-
Theorem 2 of [lo], which deals with the case k = U only. quence (OOll), let (t)' be the 4-ary spand 2 de Bruijn
Lack of space prevents detailed discussion of the method sequence
of [8]; however, we will use it only in the simple case when
k = U in Section V below.                                                        (0022013211231033),
   We now give a construction that produces PMs for most
of Etzion's parameter sets, but which gives PMs having and let (t)' be the 4-ary spand 2 de Bruijn sequence
rows with linear complexities bounded below the maxi-
 mum possible. This construction will be the first step in                       (3211220023013310).
our construction of a basic class of PMs; the other steps
will be completed in Section VI.                              Using the sequence (s) in conjunction with the sequences
   Theorem 4.4: Suppose U , U are integers with U 2 3, (t)' and (t)' in Theorem 2 of [lo] or Construction 3.2 of
                                                     '
U 2 2, and suppose further that U + 1 I 2 k I 2"- and if      [8], we obtain the following (4,16; 2,3) PMs:
k = U - 1 then U 2 3. Then there exists a (2k,2uL'-k ,U )
                                                        ;U

2uu-k - 2(~-1)V-kand whose column complexities are
                                                                                      oO01000011010110
748                                                      IEEE TRANSACTIONSON INFORMATION THEORY, VOL. 40, NO. 3, MAY 1994


and                                                                 When      U =   3 and   U =   2, we have the following parame-
                                                                  ter sets:
                       0011010001001000
                       010110111m10
               A 2 = [ 1100101110110111
                       1010010001111101
                                            j   *
                                                                      i) (22,2,; 3,2): a (4,16; 3,2) PM may be constructed
                                                                         using the method of [l]. A useful example is the
                                                                         following PM whose rows have linear complexities
                                                                         equal to 14:
It may be verified that both A, and A, have even column
sums (of linear complexity 3) and that the linear complexi-                                0011001000110111
ties of each of the rows of A, is 14. Also the rows of A,                                  0011000101100001
all have odd weight (and so their linear complexities are                                  0011011100110010
maximal).                                                0                                 1100101110011011
   Lemma 5.2. There exists (4,64; 2,4) PM A, with every              ii) (2,, 2,; 3,2): an (8,8; 3,2) PM appears in [ll], Ex-
column sum and row sum even. There exists (4,64; 2,4)                    ample 5.6. It has row complexities equal to 6 and
PM A, with every column sum even and every row sum                       column complexities equal to 5.
odd.                                                                iii) (2,, 2’; 3,2): suitable PM’s are the transpose of A,
     Prooc Let (s) be the spand 2 binary de Bruijn se-                   and A, of this section.
quence (0010, let (?), be the 4 ary spand 3 de Bruijn             Similarly when U = 3 and U = 3, we have
seauence
                                                                       i) (22,27;3,3): apply Construction 3.1 to the columns
(000202220012133201031322101312230113023211120323                         of the PM A, to obtain a PM with the desired
                                  1102123310030333)                       parameters having row complexities less than or
and let (tI4 be the 4-ary spand 3 de Bruijn sequence                      equal to 27 - 2, by Lemma 3.3.
                                                                      ii) (2,,2$3,3): a suitable PM may be obtained by
(220103132223332320012133000303310021231102032111                         applying Construction 3.1 to the rows of the
                                         2023011302210131).               (Z3,2,; 3,2) PM above and then transposing. Lemma
                                                                          3.3 guarantees that the row complexities will be less
Using the sequence (s) in conjunction with the sequences                  than or equal to Z 5 - 2,.
( t ) 3 and ( t ) , in Theorem 2 of [lo] or Construction 3.2 of      iii) (2,, 25,3,3): a suitable PM may be obtained by
[8], we obtain the following (4,64; 2,4) PM’s:                             applying Construction 3.1 to the rows of the


                           oooo1101oooo10110001ooo1010100111101011011011100110010001110
                           111100101110111011001010100oO01000010011001100010111010111111100
and

                           010OOOOOO1010010011110100oO0111011100111001101001001001111011101
                           101100101010011010001ooo1111110000010101110001100110000100101111


It may be verified that both A, and A, have even column                  (2’, 2,; 3,2) PM above and then transposing. The
sums (of linear complexity 3) and that the linear complexi-              resulting PM has column complexities 15 and row
ties of the rows of A, are all less than or equal to 57. Also            complexities less than or equal to 2’ - 2’.
the rows of A, all have odd weight (and so their linear               We claim that PMs for U = 4 and U = 5 can be con-
complexities are maximal).                                    0
                                                                   structed in a similar fashion to those above by combining
   Details of how the sequences ( t ) ’ - ( t ) 4 were obtained    the us? of Construction 3.1, Construction 3.4, and Theo-
are omitted. Linear complexities of rows were computed             rem 4.4 with the PMs A,, A,, A,, A, and the examples
using the Games-Chan algorithm described in [151.                  already obtained, while keeping track of the linear com-
   Next, we present constructions for PMs with 1 < L’ I U          plexities of rows and columns. The full details are given in
   5.             ih
 I We being wt the case U = 2 where the only possi-                the Appendix.
ble parameters are (4,4; 2,2). It is shown in [2] that, up to
cycling rows and columns, there are two possibilities:                  VI. CONSTRUCTION BASIC
                                                                                      OF A    CLASS PERFECT
                                                                                                  OF
                                                                                                     MAPS

                                                                     We are now ready to supplement the result of Theorem
                                                                   4.4 with further constructions. Our basic class is to consist
                                                                   of PMs with parameters (2k,2”‘’-k; ,U), where 6 s U <
                                                                                                        U
                                                                   2k < u2” and U I with a bound on row complexities;
                                                                                       U
 both having odd row and column sums.                              we will repeatedly apply Construction 3.1 to the rows of
PATERSON: PERFECT M A P S                                                                                              749


PMs in this class in Section VI1 and the reason for             Lemma 3.5, we obtain a (2Y-1,2U+1; PM B having
                                                                                                        u,2)
desiring such a range of parameters will become clear           minimal column complexities and row complexities equal
there.                                                          to2" + 2.                                                0
  On examining Theorem 4.4, the reader will notice that            Our next theorem deals with the case k = U in iii)
we have already constructed PMs with the desired param-         above.
eters for a large portion of the range above. Specifically,        Theorem 6.3: Suppose 1 < U I U and U 2 6. Then there
the parameter sets still missing are those with                 exists a (2"; 2""-"; U,U ) PM having row complexities less
                                                                than or equal to 2""-" - 2("-')"-".
    i) U = 1,
   ii) k = U - 1 and U = 2,                                           Proofi The proof is split into cases for U even and
  iii) k = U and U 2 2,                                         odd. We begin with U even and construct P M s having
                 +
  iv) U < k < U log, (U + 1) and      U 2   2.                  k = U for U = 6 and 1 < U I6.
                                                                   The transpose of PM A, is a (26,22;4,2) PM having
   For case i), PMs with U = 1 can easily be constructed        column complexities 57 or less. Apply Construction 3.1 to
using a (U,k ) Perfect Factor; however, we are interested       the columns of this PM twice to obtain a (26,26;6,2) PM
in PMs with low row complexities, and the simple idea of        A, having row complexities less than or equal to 26 - 2,.
ordering the sequences of such a factor as the columns of          From Section V, there exists a (2,, 26; 3,3) PM having
a PM fails to give the required complexities. It is readily     row complexities at least 2, less than the maximum. Apply
verified from Result 1.1 that for U 2 4, every PM with          Construction 3.1 three times to the rows of this PM and
U = 1 has at most two columns. It is also simple to show        transpose to obtain a (26,2"; 6,3) PM with row complexi-
that a PM with U = 1 having at most two columns must            ties less than or equal to 2l' - Z9.
have at least one odd weight row. Conversely, in the next          Again from Section V, there exists a (2,,2$3,3) PM
construction, we construct PMs for U = 1 having minimal         having row complexities at least 2, less than the maxi-
row complexities for every k IU - 2 whenever U 2 5.             mum. Applying Construction 3.1 once to the rows and
   Lemma 6.1: Suppose U 2 5 and k 2 U - 2. Then there           then three times to the columns, we obtain a (26,218;6,4)
exists a (2k,2u-k,U,1) PM having rows with linear com-          PM with row complexities less than or equal to 218 - 214.
plexity equal to 2u-k-' + 1.                                       Recall that A, is a (2', 24;2,3) PM with row complexi-
      Proofi We aim to produce a (2k,2 " - k ;U,1) PM. Then     ties equal to 14. Applying Construction 3.1 twice to the
by Result 1.1 we require 2k > U, so 2k 2 (U - 1) + 2.           rows, we obtain a (26,2,; 2,5) PM with column complexi-
Then from Lemma 4.2, there exists a (U - 1, k ) Perfect         ties less than or equal to 26 - 2,. Using Construction 3.1
Factor whose sequences have period 2k and linear com-           four times on the columns of this PM, we obtain a
plexity at most 2k - 1. Using these sequences in any            (26,zz4;6,5) PM with row complexities less than or equal
order to form the columns of an array, we obtain a              to 224 - 219.
(2k,2"-k-'; U - 1 , l ) PM A with even column sums. Ap-            Applying Construction 3.1 four times to the rows of the
plying Construction 3.1 to the columns of A, we obtain a        (26,26;6,2) PM A, above, we obtain a (2,', 26;6,6) PM
(2k,2"-k; U,1) PM B , with row complexities 2u-k-1 + 1          with column complexities less than or equal to 2,' - 2%.
according to Lemma 3.3.                                    0    Transposing, we obtain a (26,2,'; 6,6) PM with row com-
   In the case k = and k = U - 1 not covered by Lemma           plexities less than or equal to 230 - 2%.
6.1, PMs with the desired parameters may be obtained by             Still assuming U even, suppose U 2 8 and U 2 2. Write
using a de Bruijn sequence in the former case and by            U = 2s + 2, so that s 2 3. Then U - 1 2 1 and 2s 2 s +
using the sequences of a (U,U - 1) Perfect Factor as the        3, so that by Theorem 4.4 when U - 1 2 2 and by Lemma
column of an array in the latter.                               6.1 when U - 1 = 1, there exists a (2s,2(s+2xu-1)-s*  7s +

   We next address ii) above by giving a cnstruction for        2,u - 1) PM A having even row sums. Apply Construc-
PMs with k = U - 1 and U = 2 having low row complexi-           tion 3.1 to t h e rows of A to obtain a
ties.                                                           (2"+', 2(s+2xu-1)-s; + 2, U ) PM B having column com-
                                                                                      s
   Lemma 6.2: Suppose U 2 4. Then there exists a                plexities less than or equal to 2"" - 2s by Lemma 3.3.
     ',
(2"- 2"+'; U,2) PM having row complexities equal to                 For s > 1, 2s > s and so we may repeatedly apply
 2" + 2.                                                         Construction 3.1 to the columns of B a total of s times to
      Proofi Let (s) be a spand U - 2 de Bruijn sequence        obtain a (2", 2"U-k; U ) PM C having column complexi-
                                                                                      U,
having linear complexity 2"-' - 1 (such a sequence al-          ties less than or equal to 2" - 2s + s and row complexi-
ways exists; for example in [121 it is shown that a de Bruijn   ties less than or equal to 2uu-k- 2("-1)"-k.
 sequence derived from an m-sequence has this linear                This completes the cases when U is even. Next we
 complexity). Applying the operator D-' to (s), we obtain        consider U odd and U 2 7. We initially deal with the case
 a (U - 1,u - 2) Perfect Factor consisting of two se-            U = 2 and consider two subcases. First, suppose U = 7,
 quences of period and linear complexity 2"-'. Thus the          and consider the (26,2'; 4,2) PM obtained by transposing
 sequences of the Perfect Factor have odd weight.               A,. It has odd weight columns and so on applying Con-
   Use this Perfect Factor in Result 4.3 to obtain a             struction 3.4 to its columns we obtain a (Z7,2,; 5,2) PM
 (2"-',2"; U - 1,2) PM A with odd column sums. Apply-                                            +
                                                                 having column complexities 26 1. Applying Construc-
 ing Construction 3.4 to the columns of A and using              tion 3.1 twice to the columns of this PM, we obtain a
750                                                        IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 3, MAY 1994


(27,27';7 , 2 ) PM having row complexities less than or equal          In the remaining even case where U - 1 = 1, w = 1,we
to 27 - 25.                                                        use a special argument. When U = 6, consider the
    Second, suppose U 2 9. Write U = 2 s + 3 so s 2 3 and          (26,22;4,2)          PM obtained by transposing A,. It has odd
       +
2 S 2 s 4. Then by Lemma 6.1, there exists a (2', 23;s        +    weight columns and so on applying Construction 3.4 to its
3 , l ) PM A with even row sums. Apply Construction 3.1 to         columns we obtain a (27,Z3; 5 2 ) PM having column com-
the rows of A, then s times to the columns of the                                +
                                                                   plexities 26 1. Applying Construction 3.1 to the columns
                          +
resulting ( 2 S + s + 3 ,3 ;s 3 , 2 ) PM to obtain the desired
                        2                                          of this PM, we obtain a ( 2 7 , 2 5 ; 6 , 2 )PM having row
(2",2"; U ,2 ) PM having row complexities less than or             complexities less than or equal to 25 - 23. When U 2 8,
equal to 2" - 2"-'.                                                from the odd case above, there exists a ( 2 u + 1 , 2 u -,3U* -
    Last, we consider cases where U is odd and U 2 3.              1,2) PM A having even weight columns. Applying Con-
Suppose U 2 7 and U 2 3. Write U = 2 s + 1. Then U - 1             struction 3.1 to the columns of A, we obtain a
 2 2, s +                       +
             1 2 4, and 2' 2 s 2, so that by Theorem 4.4           ( 2 " + l , 2 " - l ;u , 2 ) PM with row complexities less than or
when U - 1 2 3 and by Lemma 6.2 when U - 1 = 2,                    equal to 2"-' - 2 " - 3. This completes the proof of the
                                      +
there exists a (2', 2 ( s + 1 X u - 1 ); s 1, U - 1) PM A hav-
                                       -s                          theorem.                                                         0
ing even row sums. Apply Construction 3.1 to the rows of               We summarize the results of this section in the follow-
                                           +
A to obtain a (2S+'+1,2(S+lXu-1)-3.s 1,v) PM B hav-
                                           ,                       ing:
ing column complexities at most 2 S + s + 1 2'. Now 2s > s,
                                                -                      Theorem 6.5: For every U 2 6 there exists a
so we may repeatedly apply Construction 3.1 to the                 ( 2 k ,2 U - k ; ,1) PM with row complexities equal to 2 u - k - 1
                                                                                   U
columns of B a total of s times to obtain a (2",2"U-k; ,U )
                                                          U         +                           +
                                                                        1 for every k with U 1 I 2 k I 2"-'. For every U 2 6
PM C having column complexities less than or equal to              and every 2 I U I U , there exists a ( 2 k ,2 U u - kU ,U ) PM
                                                                                                                             ;
2" - 2'  +    s and row complexities less than or equal to         with row complexities less than or equal to 2u"-k -
2uu-k - Z ( U - 1 ) ~ - k                                    0     2'" - 1 ) u - k for every k with U      +  1 I 2k < u2", i.e., with
   Our final construction in this section deals with the           [log,(u + 111 I k < U + [log,(u + 1)l.
remaining parameter sets in iv) above.
   Theorem 6.4: Suppose U 2 6 and U 2 2. Let 0 < w <                       VII. EXISTENCEPERFECTMAPSFOR EVERY
                                                                                       OF

log, (U + 1). Then there exists a (2"+",2""-"-"; U ,U ) PM                                     PARAMETER
                                                                                        POSSIBLE       SET
with row complexities less than or equal to 2""-"-" -                   In this section we use the basic class of PMs of Theo-
2'" - 1 ) u - U - w                                                  rem 6.5 to construct PMs for every possible parameter set
     Proof Again the proof is split into two cases depend- satisfying the necessary conditions of Result 1.1. In the
ing on whether U is even or odd. We suppose first that U following theorem we study the problem of general U and
is odd, so U 2 7 and v 2 2. Write U = 2s + 1, so that U with U 2 6, 1 < U I U. Then we combine these studies
s 2 3, and suppose 0 < w < log,(u + 1). It is easy to with Lemma 6.1 and the results of Section V to produce
verify that 2' 2 s + w + 1 and 2 ( s + w + 1 X u - 1 > s - 1. our last result.
                                                             )- U
By Theorem 4.4 or Lemma 6.1 in the case U = 2, there                    Theorem 7.1. Suppose U 2 6 and 1 < v IU. Then
exists a (2', ~ ( s + w + ~ X U - ~ ) ;Ss + w + 1,v - 1 ) P M A hav- there exist ( 2 k , 2 u v - k ; ~ , ~ ) for every set of parame-
                                      -                                                               PMs
ing even row sums.                                                   ters satisfying Result 1.1.
   Applying Construction 3.1 to the rows of A, we obtain                  Proof Suppose U 2 6 and 1 < U IU. From Result
                                      +
a (22S+W+l,2(S+w+lXu-l)-s.s + w 1, U ) PM B with col- 1.1we see that we must construct ( 2 k ,2""-k;U ,U ) PMs for
                                      ,
umn complexities less than or equal to 2 2 s + w + 1 2'. For every k with U < 2k and 2uu-k > U . Writing k = w + ut
                                                            -
s 2 3, 2' > s and so we may apply Construction 3.1 s - w where [log, (U + 111 I w < U + [log, (U + 111 and 0 I t
times to the columns of B to obtain a (2"+",2""-"-"; U ,U ) - U - 1, we split the proof into two cases: 0 I t I U - 2
                                                                      <
PM having row complexities less than or equal to 2""-"-" a n d t = u - 1 .
 - 2(u-l)u-u-w . Also, since we make strictly less than 2 S             Suppose 0 It I U - 2, so 2 I U - t I U . Then from
applications of Construction 3.1, we can deduce that this our choice for the range of w , 2w > U. Moreover
PM has even weight columns.                                                 u ( u - t ) - w 2 2 u - U - [log,(u + 1)l + 1
   Suppose now that U is even, U 2 6, and U 2 2. Write
                                                                                            = U + 1 - [log, (U + 111,
U = 2 s and suppose 0 < w < log,(u + 1). Then s 2 3,
2 S 2 s + w , and 2 ( s + w X u - 1 ) -> U - 1. By Theorem 4.4,
                                       s                            and for   U 2   6
or Lemma 6.1 in the case U - 1 = 1, w > 1, and Lemma                                              >
                                                                                    2u+1-rl0g2(~+l)i                    - t.
6.2 in the case U - 1 = 2, w = 1, there exists a
                     +
(2S, 2(s+ZXu-l)-s;s w , U - 1) PM A with even row sums.             Then by Theorem 6.5 there exists a (2", 2 u ( u - t ) - ,U-,U - t )
                                                                                                                            w
   Applying Construction 3.1 to the rows of A, we obtain            PM A with row complexities less than or equal to
a (22S+2,2(s+W)(u-l)-s. + w , U ) PM B with column com-
                         ,s                                         2u(u-')-w - 2(u-lXu-t)-w . Now
plexities less than or equal to 2"+" - 2'. For s 2 3,               (U -   l)(u - t )   -   w 2 2 ( u - 1) - U - Ilog,(u            +01+1
2' > s and so we may apply Construction 3.1 s - w times                                       = U - 1 - [log, (U + 1)1,
to the columns of B to obtain a (2"+",2uu-u-";U,U ) PM
having row complexities less than or equal to 2""-"-" -             and for   U 26
2("-1)u-u-w                                                                    2"- i - r i o g , ( u + i ) l 2 u - 2 2v - 2 2 1 .
PATERSON: PERFECT M A P S                                                                                                                                        75 1


Hence the row complexities of A are at least t below the                                           gous to the result in [91 for binary de Bruijn sequences.
maximum possible value and so we may make t applica-                                               This number is known in the simplest case: there are two
tions of Construction 3.1 to the rows of A to obtain a                                             PM's with parameters (4,4; 2,2)-details may be found in
( 2 W + u I , 2 U ( u - ' ) - W ; U, U ) PM, i.e., a ( 2 W + U I , 2 U U - ( w + U I ) .
                                                                                         ,U, U )   PI.
PM. (Note that when t = 0 there is nothing to be done; A                                              Higher-dimensional arrays analogous to Perfect Maps
is a suitable PM.)                                                                                 and de Bruijn sequences can be defined. There are some
     In the second case we suppose t = U - 1 and we may                                            existence results for the three-dimensional generalization
assume that [log,(u + 1)1 Iw IU. We aim to produce a                                               in [16]. A complete solution would be of doubtful practical
PM having 2"-" columns and so from Result 1.1, we                                                  significance, but mathematically interesting.
require
                                                                                     APPENDIX
                       2"-" 2 U 1.                  +
                                                    (7.1)
                                                            In this appendix we give constructions for PMs with 1 < U I U
When w = U or w = U - 1, this inequality is not satisfied for U = 4 and U = 5, these being the parameter sets omitted in
(since U > 11, and so no such PM can exist. We may Section V.
therefore assume that U - w 2 2 and so by Theorem 6.5
there exists a (2", 2"-"; U,1) PM A with row complexities A. The case U = 4
equal to 2'-"-' + 1. We wish to make t = U - 1 applica-     1) The case U = 4, U = 2
tions of Construction 3.1 to the rows of A, so we require  i) (23,25;4,2): apply Construction 3.1 to the columns of the
                     2u-"      -   (2"-" - I      + 1) 2 U - 1,                                          (23,23;3,2) PM of Section V to obtain a PM with the
                                                                                                         desired parameters having columns with linear complexities
i.e.,                                                                                                    6 and rows with linear complexities less than or equal to
                                     2u-w-1
                                                  (7.2)
                                                   2   U.                                                25 - 23,
                                                                                                   ii) (z4,2,;4,2): apply Construction 3.1 to the rows of the PM
If 2"-" 2 2u, then (7.2) holds and we proceed to make t                                                A, of Section V and transpose to obtain a PM with the
applications of Construction 3.1 to the rows of A to                                                   desired parameters having columns with linear complexities
obtain the required (2"'-u+w, 2'-"; U,U ) PM. Otherwise,                                               15 and rows with linear complexities less than or equal to
from (7.1) we have                                                                                       24 - 22.
                                                                                                   iii) (25,23;4,2): apply Construction 3.4 to the rows of the PM
             U + 1 I    2"-" < 2 u I 2" for U > 1,
                                                                                                        A , of Section V and transpose to obtain a PM with the
and the desired ( 2 u u - u + w , 2 u - w ; ~ , ~ )may be ob-
                                             PM                                                         desired parameters having columns with linear complexities
tained from the transpose of a PM from Result 4.3.                                                         +
                                                                                                        Z4 1 and rows with linear complexities 22 2.+
    This completes the proof for t = U - 1, the second of                                          iv) (26,22;4,2): the PMs A, and A , of Section V may be
our two cases.                                              0                                           transposed to obtain PMs with the desired parameters with
     Theorem 7.2: The conditions of Result 1.1 on U, U , and                                            columns with linear complexities either less than or equal to
r = 2 k , s = 2 u u - k are sufficient for the existence of a                                           57 or maximal and with rows with linear complexities equal
                                                                                                        to 3.
( 2 k , 2uu-k U, U ) Perfect Map.
             ;
                                                                                                      2) The case U = 4, U = 3
        Pro05 Suppose without loss of generality that U 2 U.
Construction of Perfect Maps for U = 1 and every possi-                                              i) (23,29;4,3): apply Construction 3.1 to the columns of the
ble r = 2 k for each U were given in Lemma 6.1 so we may                                                (23,26;3,3)PM of Section V to obtain a PM with the
assume that U > 1. When U I5 the results of Section V                                                   desired parameters having columns with linear complexities
show the existence of PM's for each U IU and every                                                      8 and rows with linear complexities less than or equal to
r = 2 k , s = 2 u u - k satisfymg the conditions of Result 1.1.                                           29 - 26.
                                                                                                    ii) (Z4,28;4,3): apply Construction 3.1 to the columns of the
When U 2 6 and U > 1, Theorem 7.1 may be used to                                                        (24,25;3,3)PM of Section V to obtain a PM with the
construct PM's for every r = 2 k , s = 2uu-k satisfying the                                             desired parameters having columns with linear complexities
conditions of Result 1.1.                                   0                                           16 and rows with linear complexities less than or equal to
                                                                                                          28 - 25.
             VIII. RELATED   OPEN  PROBLEMS
                                                                                                    iii) (2s,27;4,3): apply Construction 3.1 to the rows of the
  It has already been noted that Perfect Maps are the                                                    (2,, 25;3,3) PM of Section V and transpose to obtain a PM
two-dimensional generalizations of binary de Bruijn se-                                                  with the desired parameters having columns with linear
quences. The existence (and indeed number) of such                                                                                                      +
                                                                                                         complexities less than or equal to Z5 - 22 1 and rows
sequences has long been known, even in the nonbinary                                                     with linear complexities less than or equal to 27 - 2,.
case. We can ask for which parameter sets do Perfect                                                iv) (26,26;4,3): apply Construction 3.1 to the rows of the
Maps exist in the more general c-ary case? Necessary                                                     (23,26;3,3) PM of Section V and transpose to obtain a PM
                                                                                                         with the desired parameters having columns with linear
conditions similar to those of Result 1.1 can be obtained.                                                                                              +
                                                                                                         complexities less than or equal to 26 - Z3 1 and rows
We conjecture that these are also sufficient. Also, we have                                              with linear complexities less than or equal to 26 - 23.
constructed many Perfect Maps for each parameter set                                                 v) (2',Z5;4,3): apply Construction 3.1 to the rows of the PM
(we do not attempt to enumerate how many here); it                                                       of A.li). The resulting PM has columns with linear com-
would be of interest to give an exact expression for the                                                 plexities less than or equal to 2' - Z3 and rows with linear
number of Perfect Maps for each parameter set, analo-                                                                                               +
                                                                                                         complexities less than or equal to 25 - Z3 1.
752                                                           IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 3, MAY 1994


 vi) (28, Z4;4,3): apply Construction 3.1 to the rows of the PM              or equal to 29 - P.
      of A.lii). The resulting PM has columns with linear com-           v) (27,2’; 53): apply Construction 3.1 to the columns of the
      plexities less than or equal to 2’ - 24 and rows with linear           PM of A.2v) to obtain a PM with the required parameters
      complexities less than or equal to 13.                                 having columns with linear complexities less than or equal
 vii) (29,23; 4,3): apply Construction 3.1 to the rows of the PM                          +
                                                                             to 27 - Z4 2 and rows with linear complexities less than
      of kliii). The resulting PM has columns with lienar com-               or equal to 2’ - 2’.
      plexities less than or equal to 29 - 2’ and rows with linear      vi) PM’s for U = 5 and U = 3 with 8 I k I having rows with
                                                                                                                    12
      complexities equal to 7.                                               even weights may be obtained by applying Construction 3.1
viii) (21°, 2’; 4,3): apply Construction 3.1 to the rows of the PM           to the rows of the corresponding (2k-5,215-k; 2 ) PMs in
                                                                                                                            5
      of A.liv) having even row sums. The resulting PM has                   B.l) above.
      columns with linear complexities less than or equal to            vii) (213,2’; 5,3): such a PM may be obtained from Result 4.3.
      2” - 26 and rows with linear complexities less than or
      equal to 58.                                                        3) The case U   =   5, U = 4
   3) The case U = 4, U = 4                                                i) (23, 217;5,4): a PM with these parameters having rows with
  PMs for   U = 4 and U = 4 with 3 I I 8 having rows with
                                        k                                     linear complexities less than or equal to 217 - 213 may be
linear complexities less than or equal to 216-k - 212-k may be                obtaned from Theorem 4.4.
obtained by applying Construction 3.1 to the rows of the corre-           ii) (Z4, 216;5 , 4 ) a PM with these parameters having rows with
sponding (2”-‘, 2k;4,3) PMs in A.2) above and transposing.                    linear complexities less than or equal to 216 - 2” may be
The resulting PMs have columns with linear complexities one                   obtained from Theorem 4.4.
greater than the linear complexities of the rows of the corre-          iii)c5,    2”; $4): apply Construction 3.1 to the columns of the
sponding PMs of A.2).                                                         (2’, 2”; 4,4) PM of A.3) to obtain a PM with the required
                                                                              parameters having columns with linear complexities less
B. The case U   =   5                                                         than or equal to 25 - 2, + 3 and rows with linear complexi-
  I) The case U     =   5,   U =   2                                          ties less than or equal to 215 - 211.
  i) (23, Z7;5 2 ) : apply Construction 3.1 to the columns of the        iv) (26, 214; 5 4 ) : apply Construction 3.1 to the columns of the
     PM of A.li). The resulting PM has rows with linear complex-              (26,2”; 4,4) PM of A.3) to obtain a PM with the required
     ities less than or equal to Z7 - 2’.                                     parameters having linear complexities less than or equal to
 ii) (24, 26; 5,2): apply Construction 3.1 to the columns of the                     +
                                                                              26 - Z3 2 and rows with linear complexities less than or
     PM of A X ) . The resulting PM has rows with linear com-                 equal to 214 - 2”.
     plexities less than or equal to 26 - Z4.                             v) (27,213; 54): apply Construction 3.1 to the columns of the
iii) (2’,2’;5,2): apply Construction 3.1 to the columns of the                (Z7, 2’; 4,4) PM of A.3) to obtain a PM with the required
     PM of A.liii). The resulting PM has rows with linear com-                parameters having columns with linear complexities less
     plexities less than or equal to 2’ - 2, and columns with                 than or equal to 27 - 2,   +  2 and rows with linear complexi-
     linear complexities 24 2.     +                                          ties less than or equal to 213 - Z9.
iv) (26, Z4; 5,2): apply Construction 3.1 to the rows of the PM         vi) PMs for U = 5 and U = 4 with 8 I I12 having rows with
                                                                                                                    k
     A, of Section V and transpose to obtain a PM with the                    even weight may be obtained by applying Construction 3.1
     desired parameters and rows with linear complexities less                to the rows of the corresponding ( 2 k - 5 ,220-k;5 3 ) PMs in
     than or equal to 2, - 2’.                                                B.2) above.
  v) (2’, Z3; 5 , 2 ) apply Construction 3.4 to the rows of the PM      vii) Similarly, PMs for U = 5 and U = 4 with 13 I k 5 17 having
     A, of Section V and transpose to obtain a PM with the                    rows with even weight (except when k = 17) may be ob-
     desired parameters having columns with linear complexities               tained by applying Construction 3.1 twice to the rows of the
                  +
     equal to 26 1 and rows with linear complexities equal to 6.              corresponding (2k- lo,220-k;   5,2) PMs in B.l) above.
 vi) (2’, 22; 5,2): such a PM may be obtained by transposing a
                                                                          4) The case   U = 5, U =   5
     (22, 2’; 2,5) PM from Result 4.3.
                                                                          i) (2,, 222;5 5 ) : a PM with these parameters may be obtained
  2) The case U = 5, L- = 3
                                                                             from Result 4.3.
    i) (23,212;5,3): a PM with these parameters and rows with            ii) (Z4,2’l; 5 5 ) : a PM with these parameters may be obtained
       linear complexities less than or equal to 212 - 29 may be             from Result 4.3.
       derived from Theorem 4.4.                                        iii) (2’, 2”; 5 , 5 ) a PM with these parameters may be obtained
   ii) (2,, 211;5,3): apply Construction 3.1 to the columns of the           from Result 4.3.
       PM of A.2ii) to obtain a PM with the required parameters         iv) (26,219;5,5): apply Construction 3.1 to the rows of the
       having rows with linear complexities less than or equal to            (214,26; 5,4) PM of B.3) and transpose to obtain a PM with
       2“ - 28.                                                              the required parameters.
 iii) (2’, 21°; 5 3 ) : apply Construction 3.1 to the columns of the
                                                                         V (z7,2l8;5,5): apply Construction 3.1 to the rows of the
                                                                           )
       PM of A2iii) to obtain a PM with the required parameters              (213, 27;5,4) PM of B.3) and transpose to obtain a PM with
       having columns with linear complexities less than or equal            the required parameters.
       to 25 - 2’   +   2 and rows with linear complexities less than
                                                                        vi) PMs for U = U = 5 with 8 I I 12 may be obtained by
                                                                                                              k
       or equal to 21° - 27.
                                                                             applying Construction 3.1 to the rows of the corresponding
  iv) (26,29;5,3): apply Construction 3.1 to the columns of the
                                                                             (2k-5, 225-k;5 4 ) PMs in B.3) above.
       PM of A2iv) to obtain a PM with the required parameters
       having columns with linear complexities less than or equal         This completes the constructions required to prove the exis-
                    +
       to 26 - 23 2 and rows with linear complexities less than         tence of PMs with 1 < U IU for U = 4 and U = 5. Note that we
PATERSON: PERFECX M A P S                                                                                                                      753


have constructed PMs with rows and columns having linear                   [6] E. M. Petriu, J. S . Basran, and F. C. A. Groen, “Automated guided
complexities often considerably below the maximal values.                      vehicle position recovery,” IEEE Trans, Instrum. Meas., vol. 39, pp.
                                                                               254-258. 1990.
                      ACKNOWLEDGMENTS                                      [7] B. Arazi, “Position recovery using binary sequences,” EIectron.
                                                                               Len., vol. 20, pp. 61-62, 1984.
  The author is indebted to Chris Mitchell for much                        [8] C. J. Mitchell-and K. G. Paterson, “Decoding Perfect Maps,”
                                                                               Designs, Codes Cryprogr.,vol. 4, no. 1 (1994) pp. 11-30.
encouragement and advice and to Fred Piper, Peter Wild,                    [91 N. G. de Bruijn, ‘‘A combinatorial problem,,,in proc, Nederlansh
and Matt Robshaw for many valuable discussions. The                            Akademie van Wetenschappen,vol. 49, pp. 758-764, 1946.
thorough comments of the referees on an earier version                    [lo1 S . L. Ma, “A note on binaj arrays witha certain window property,”
                                                                               IEEE Trans. Inform. Theory, vol. IT-30, pp. 774-775, 1984.
of this work are gratefully acknowledged.                                 1111 C. T. Fan. S. M. Fan. S. L. Ma. and M. K. Siu, “On de Bruiin
                                                                          ~~




                                                                               arrays,” A& Combinabria, vol. 1 9 4 pp. 205-213,’1985.
                           REFERENCES                                     [12] A. H. Chan, R. A. Games, and E. L. Key, “On the complexities of
     T. Etzion, “Constructions for perfect maps and pseudo-random              de Bruijn sequences,”J. Comb. Theoy, Ser. A, vol. 33, pp. 233-246,
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     1988.                                                                [13] A. Lempel, “On a homomorphism of the de Bruijn graph and its
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     Oxford: Clarendan Press 1993.                                             705-709, 1984.
     J. A. Bondy and U. S . R. Murty, “Graph Theoly with Applications.    [15] R. A. Games and A. H. Chan, “A fast algorithm for determining
     New York Elsevier, 1976.                                                  the complexity of a binary sequence with period 2”,” IEEE Trans.
     E. M. Petriu and J. S . Basran, “On the position measurement of           Inform. Theory, vol. IT-29, pp. 144-146, 1983.
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     Trans. Instrum.Meas., vol. 38, pp. 799-803, 1989.                         Ars Combinatoria, vol. 29C, pp. 33-40, 1990.

				
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