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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, arrays,” IEEE Trans. Inform. &oy, vol. IT-34, pp. 1308-1316, 1982. 1988. [13] A. Lempel, “On a homomorphism of the de Bruijn graph and its I. S . Reed and R. M. Stewart, “Note on the existence of Perfect applicationsto the design of feedback shift registers,” IEEE Trans. Maps,” I R E Trans. Inform. Theoy, vol. IT-8, pp. 10-12, 1962. Comput., vol. C-19, pp. 1204-1209, 1970. J. Bums and C. J. Mitchell, “Coding schemes for two-dimensional [I41 T. Etzion and A. Lempel, “Construction of de Bmijn sequences of position sensing,’’ in Cryptography and Coding 111, M. Ganley, Ed. minimal complexity,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 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. automated guided vehicles using pseudorandom encoding,” IEEE [16] A. M. Ivlnyi, “Constructionof three-dimensionalperfect matrices,” Trans. Instrum.Meas., vol. 38, pp. 799-803, 1989. Ars Combinatoria, vol. 29C, pp. 33-40, 1990.

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