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Computer Construction of Quasi-Twisted Two-Weight Codes

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

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Computer Construction of Quasi-Twisted Two-Weight Codes

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  1. Computer Construction of Quasi-Twisted Two-Weight Codes Eric Chen eric.chen@hkr.se Dept. of Comp.Science Kristianstad University 29188 Kristianstad Sweden

  2. 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.

  3. Outline • Two-weight codes and graphs • Cyclic codes • Quasi-twisted (QT) codes • QT simplex codes • Construction of QT two-weight codes • Results

  4. 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 projectiveif any two of its coordinates are linearly independent, or, if the minimum distance of its dual code is at least three.

  5. 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.

  6. 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

  7. 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

  8. Quasi-Twisted (QT) Codes • a consta-cyclic shift of any codeword by ppositions 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

  9. 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

  10. 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.

  11. Consta-Cyclic Simplex[21, 3]4Code twistulant generator matrix

  12. Quasi-Twisted Simplex[21, 3]4Code QT form of generator matrix

  13. Quasi-Twisted Simplex[21, 3]4Code 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

  14. Weight Matrix • Weight matrix for A(x) • It is cyclic • Example

  15. 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.

  16. 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.

  17. 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

  18. 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

  19. Database of 2-Weight Codes • http://www.hkr.se/ ~chen/research/2-weight-codes/search.php

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