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Hermitian self-dual quasi-abelian codes

Yıl 2018, , 5 - 18, 15.01.2018
https://doi.org/10.13069/jacodesmath.327399

Öz

Quasi-abelian codes constitute an important class of linear codes containing theoretically and practically interesting codes such as quasi-cyclic codes, abelian codes, and cyclic codes. In particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes. Based on the well-known decomposition of quasi-abelian codes, the characterization and enumeration of Hermitian self-dual quasi-abelian codes are given. In the case of 1-generator quasi-abelian codes, we offer necessary and sufficient conditions for such codes to be Hermitian self-dual and give a formula for the number of these codes. In the case where the underlying groups are some $p$-groups, the actual number of resulting Hermitian self-dual quasi-abelian codes are determined.

Kaynakça

  • [1] L. M. J. Bazzi, S. K. Mitter, Some randomized code constructions from group actions, IEEE Trans. Inform. Theory 52(7) (2006) 3210–3219.
  • [2] J. Conan, G. Séguin, Structural properties and enumeration of quasi–cylic codes, Appl. Algebra Engrg. Comm. Comput. 4(1) (1993) 25–39.
  • [3] B. K. Dey, On existence of good self–dual quasicyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1794–1798.
  • [4] B. K. Dey, B. S. Rajan, Codes closed under arbitrary abelian group of permutations, SIAM J. Discrete Math. 18(1) (2004) 1–18.
  • [5] C. Ding, D. R. Kohel, S. Ling, Split group codes, IEEE Trans. Inform. Theory 46(2) (2000) 485–495.
  • [6] S. Jitman, S. Ling, Quasi–abelian codes, Des. Codes Cryptogr. 74(3) (2015) 511–531.
  • [7] S. Jitman, S. Ling, P. Solé, Hermitian self–dual abelian codes, IEEE Trans. Inform. Theory 60(3) (2014) 1496–1507.
  • [8] A. Ketkar, A. Klappenecker, S. Kumar, P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory 52(11) (2006) 4892–4914.
  • [9] K. Lally, P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math. 111(1–2) (2001) 157–175.
  • [10] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47(7) (2001) 2751–2760.
  • [11] S. Ling, P. Solé, Good self–dual quasi–cyclic codes exist, IEEE Trans. Inform. Theory 49(4) (2003) 1052–1053.
  • [12] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes III: Generator theory, IEEE Trans. Inform. Theory 51(7) (2005) 2692–2700.
  • [13] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics 17, Springer–Verlag, Berlin, Heidelberg, 2006.
  • [14] J. Pei, X. Zhang, 1-generator quasi–cyclic codes, J. Syst. Sci. Complex. 20(4) (2007) 554–561.
  • [15] V. Pless, On the uniqueness of the Golay codes, J. Combinatorial Theory 5(3) (1968) 215–228.
  • [16] B. S. Rajan, M. U. Siddiqi, Transform domain characterization of abelian codes, IEEE Trans. Inform. Theory 38(6) (1992) 1817–1821.
  • [17] G. Séguin, A class of 1-generator quasi–cyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1745–1753.
  • [18] S. K. Wasan, Quasi abelian codes, Publ. Inst. Math. 21(35) (1977) 201–206.
Yıl 2018, , 5 - 18, 15.01.2018
https://doi.org/10.13069/jacodesmath.327399

Öz

Kaynakça

  • [1] L. M. J. Bazzi, S. K. Mitter, Some randomized code constructions from group actions, IEEE Trans. Inform. Theory 52(7) (2006) 3210–3219.
  • [2] J. Conan, G. Séguin, Structural properties and enumeration of quasi–cylic codes, Appl. Algebra Engrg. Comm. Comput. 4(1) (1993) 25–39.
  • [3] B. K. Dey, On existence of good self–dual quasicyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1794–1798.
  • [4] B. K. Dey, B. S. Rajan, Codes closed under arbitrary abelian group of permutations, SIAM J. Discrete Math. 18(1) (2004) 1–18.
  • [5] C. Ding, D. R. Kohel, S. Ling, Split group codes, IEEE Trans. Inform. Theory 46(2) (2000) 485–495.
  • [6] S. Jitman, S. Ling, Quasi–abelian codes, Des. Codes Cryptogr. 74(3) (2015) 511–531.
  • [7] S. Jitman, S. Ling, P. Solé, Hermitian self–dual abelian codes, IEEE Trans. Inform. Theory 60(3) (2014) 1496–1507.
  • [8] A. Ketkar, A. Klappenecker, S. Kumar, P. K. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory 52(11) (2006) 4892–4914.
  • [9] K. Lally, P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math. 111(1–2) (2001) 157–175.
  • [10] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47(7) (2001) 2751–2760.
  • [11] S. Ling, P. Solé, Good self–dual quasi–cyclic codes exist, IEEE Trans. Inform. Theory 49(4) (2003) 1052–1053.
  • [12] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes III: Generator theory, IEEE Trans. Inform. Theory 51(7) (2005) 2692–2700.
  • [13] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics 17, Springer–Verlag, Berlin, Heidelberg, 2006.
  • [14] J. Pei, X. Zhang, 1-generator quasi–cyclic codes, J. Syst. Sci. Complex. 20(4) (2007) 554–561.
  • [15] V. Pless, On the uniqueness of the Golay codes, J. Combinatorial Theory 5(3) (1968) 215–228.
  • [16] B. S. Rajan, M. U. Siddiqi, Transform domain characterization of abelian codes, IEEE Trans. Inform. Theory 38(6) (1992) 1817–1821.
  • [17] G. Séguin, A class of 1-generator quasi–cyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1745–1753.
  • [18] S. K. Wasan, Quasi abelian codes, Publ. Inst. Math. 21(35) (1977) 201–206.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Herbert S. Palines Bu kişi benim

Somphong Jitman 0000-0003-1076-0866

Romar B. Dela Cruz Bu kişi benim

Yayımlanma Tarihi 15 Ocak 2018
Yayımlandığı Sayı Yıl 2018

Kaynak Göster

APA Palines, H. . S., Jitman, S., & Cruz, R. B. D. (2018). Hermitian self-dual quasi-abelian codes. Journal of Algebra Combinatorics Discrete Structures and Applications, 5(1), 5-18. https://doi.org/10.13069/jacodesmath.327399
AMA Palines HS, Jitman S, Cruz RBD. Hermitian self-dual quasi-abelian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. Ocak 2018;5(1):5-18. doi:10.13069/jacodesmath.327399
Chicago Palines, Herbert S., Somphong Jitman, ve Romar B. Dela Cruz. “Hermitian Self-Dual Quasi-Abelian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 5, sy. 1 (Ocak 2018): 5-18. https://doi.org/10.13069/jacodesmath.327399.
EndNote Palines HS, Jitman S, Cruz RBD (01 Ocak 2018) Hermitian self-dual quasi-abelian codes. Journal of Algebra Combinatorics Discrete Structures and Applications 5 1 5–18.
IEEE H. . S. Palines, S. Jitman, ve R. B. D. Cruz, “Hermitian self-dual quasi-abelian codes”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 5, sy. 1, ss. 5–18, 2018, doi: 10.13069/jacodesmath.327399.
ISNAD Palines, Herbert S. vd. “Hermitian Self-Dual Quasi-Abelian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 5/1 (Ocak 2018), 5-18. https://doi.org/10.13069/jacodesmath.327399.
JAMA Palines HS, Jitman S, Cruz RBD. Hermitian self-dual quasi-abelian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5:5–18.
MLA Palines, Herbert S. vd. “Hermitian Self-Dual Quasi-Abelian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 5, sy. 1, 2018, ss. 5-18, doi:10.13069/jacodesmath.327399.
Vancouver Palines HS, Jitman S, Cruz RBD. Hermitian self-dual quasi-abelian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018;5(1):5-18.

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