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					Computer Construction of
    Quasi-Twisted
  Two-Weight Codes


           Eric Chen
      eric.chen@hkr.se
    Dept. of Comp.Science
    Kristianstad University
     29188 Kristianstad
            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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