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```					Computer Construction of
Quasi-Twisted
Two-Weight Codes

Eric Chen
eric.chen@hkr.se
Dept. of Comp.Science
Sweden
Main Results

•   Computer construction of
quasi-twisted 2-weight codes
•   Many 2-weight codes can be
constructed as quasi-twisted
(QT) codes.
•   Some new QT 2-weigth codes
are obtained.
Outline

•   Two-weight codes and graphs
•   Cyclic codes
•   Quasi-twisted (QT) codes
•   QT simplex codes
•   Construction of QT two-
weight codes
•   Results
Two-Weight Codes

•    A q-ary [n, k] code is a two-
weight code if any non-zero
codeword has a weight of w1
or w2.
•    Notation: [n, k; w1, w2]q code
•    Projective code
– A code is said to be projective if
any two of its coordinates are
linearly independent, or, if the
minimum distance of its dual code is
at least three.
Strongly Regular Graphs

•       A graph with v vertices and degree k
is strongly regular if there are also
integers λ and μ such that:
–     Every two adjacent vertices have λ
common neighbours.
–     Every two non-adjacent vertices have μ
common neighbours.
•       A graph of this kind is sometimes
said to be an srg(v,k,λ,μ).
•       Projective two-weight codes are
closely related to strongly regular
graphs.
Cyclic Codes

•    q-ary linear [n, k]q code:
– n: block length
– k: code dimension
•    Cyclic [n, k]q code:
– Any codeword shifted by 1
position is still a codeword
– generator polynomial g(x)
– Generator matrix G
•   A cyclic matrix
λ-Consta-Cyclic Codes

•    for any codeword (a0, a1, ...,
an-1), a consta-cyclic shift by
one position or (λ an-1, a1, ...,
an-2), is also a codeword
– Where λ is non-zero element of
GF(q)
•    The generator matrix of an λ-
consta-cyclic code can be an
λ-consta-cyclic matrix
•    A cyclic code is an λ-consta-
cyclic code with λ = 1
Quasi-Twisted (QT) Codes

•    a consta-cyclic shift of any
codeword by p positions is
still a codeword.
•    The generator matrix of a QT
code can be written as rows of
p consta-cyclic matrices
(twistulant matrices)
•    a consta-cyclic code is a QT
code with p = 1,
•    a quasi-cyclic (QC) code is a
QT code with λ = 1
Simplex Codes

•    Simplex [(qt–1)/(q–1), t]q code
– equi-distance code, d = qt-1
– All non-zero codewords have the
same weight, d = qt-1
•    A λ-consta-cyclic simplex
code can be defined by a
generator polynomial g(x) =
(xn–l)/h(x),
– where n=(qt–1) /(q–1), and λ is a
non-zero element of GF(q) and
has order of q–1
QT Simplex Codes

•       If n=(qt–1) /(q–1) = mr, Simplex
[(qt–1)/(q–1), t]q code can be put
into QT from.
•       Example:simplex [21, 3]4 code
–     n = 21 = mp = 3 × 7, m = 3, p = r, q
= 4.
–     Let 0, 1, a, and b = 1 + a be
elements of GF(4),
–     λ=b. Then a λ-consta-cyclic matrix
defined by c(x) = 1+ bx + bx3 + bx4
+ bx5 + ax6 +x7 + x8 + ax9 + x10 +
ax11 + x13 +ax15 +bx16 +x17 + x18.
Consta-Cyclic
Simplex [21, 3]4 Code
twistulant generator matrix
Quasi-Twisted
Simplex [21, 3]4 Code
QT form of generator matrix

1   1   0      1  0                1   1     1  0   0  1 0
0   1   1   1    1  0                  1      0    0 0  1
   0   1      1  1  0                    1   0 0     0 

                                                                         
0      1   1   1   0      1      0          1 1     1      0   0
   0      0   1   1   1    1      0          1        0      0
1      0      0   1      1      1  0          1       0   0   
                                                                         
0      0   0      1   1   1   0    1        0    1   1     1
0   0       0        0   1   1   1  1        0      1     
   0   0   1      0      0   1    1        1  0        1  
                                                                      
         0      0   0      1   1   1   0    1  0        1 1
A  1         0   0       0        0   1   1   1  1  0        1 .
   1         0   0   1      0      0   1    1  1    0     
                                                                      
    1              0      0   0      1   1 1 0  1      0  

         1         0   0       0        0 1 1 1    1    0 
            1         0   0   1  0        0 1  1      1  0
   0          1              0  0       0  1 1 1   0    1 

                                                                      
1    0          1         0 0             0  0 1   1   1  1
0   1                  1   0 0           1  0  0   1    1 
1    1        0          1              0  0 0    1   1   1   0
                                                                         
   1      1    0          1             0 0   0      0   1   1

     1     0   1            1             0 0 1    0      0   1

Quasi-Twisted
Simplex [21, 3]4 Code
QT form of generator matrix
– Representation by polynomials
•    a1(x) = 1 +x, a2(x) = b + ax + x2 ,
a3(x) = ax + bx2 , a4(x) = b + x + x2,
a5(x) = b + ax + x2, a6(x) = b,
a7(x) = a+ x. r = 7
Weight Matrix

•    Weight matrix for A(x)
– It is cyclic
•    Example
Computer Construction of
QT 2-Weight Codes
•    Given a simplex [mr, t]q code of
composite length
– n =(qt–1) /(q–1) = mr
•    Find the generator polynomial,
•    Obtain A(x) and weight matrix
•    To construct a QT 2-weight [mp,
t; w1, w2] code, it is to find p
columns such that the row sums
of the selected columns give w1
or w2.
Computer Construction of
QT 2-Weight Codes
•    Example
– From simplex [21, 3]4 code with
m=3
– A QT 2-weight [9, 3; 6, 8]4 code
can be constructed by columns 1, 2,
and 4.
Results

•    A large amount of QT 2-weight
codes have been obtained.
•    Most codes have the same
parameters as known codes.
•    They may not be equivalent
– Exmaple [154, 6; 99, 108]3 code
•   Gulliver constructed with m = 11, p =14
•   Using the method above, m = 7, p =22
•   They are not equivalent
•    Some new codes are obtained
Some New Codes

•    Ternary QT 2-weight codes
– m = 671, k = 10
•   [6710, 10; 4455, 4536] code
•   [8052, 10; 5346, 5427] code
– m = 3796, k = 12
•   [7592, 12; 5022, 5103] code
– m = 7592, k = 12
•   [129064, 12; 86022, 86751] code
•    Other codes
– [595, 4; 546, 559]13 code
– [1785, 4; 1638, 1651]13 code
Database of 2-Weight Codes

•   http://www.hkr.se/
~chen/research/2-weight-codes/search.php

```
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