Hermitian self-dual quasi-abelian codes

Abstract

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.

Keywords

• Hermitian self-dual codes,Quasi-abelian codes,1-generator,p-groups

References

1. [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. [2] J. Conan, G. Séguin, Structural properties and enumeration of quasi–cylic codes, Appl. Algebra Engrg. Comm. Comput. 4(1) (1993) 25–39.
3. [3] B. K. Dey, On existence of good self–dual quasicyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1794–1798.
4. [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. [5] C. Ding, D. R. Kohel, S. Ling, Split group codes, IEEE Trans. Inform. Theory 46(2) (2000) 485–495.
6. [6] S. Jitman, S. Ling, Quasi–abelian codes, Des. Codes Cryptogr. 74(3) (2015) 511–531.
7. [7] S. Jitman, S. Ling, P. Solé, Hermitian self–dual abelian codes, IEEE Trans. Inform. Theory 60(3) (2014) 1496–1507.
8. [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. [9] K. Lally, P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math. 111(1–2) (2001) 157–175.
10. [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. [11] S. Ling, P. Solé, Good self–dual quasi–cyclic codes exist, IEEE Trans. Inform. Theory 49(4) (2003) 1052–1053.
12. [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. [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. [14] J. Pei, X. Zhang, 1-generator quasi–cyclic codes, J. Syst. Sci. Complex. 20(4) (2007) 554–561.
15. [15] V. Pless, On the uniqueness of the Golay codes, J. Combinatorial Theory 5(3) (1968) 215–228.
16. [16] B. S. Rajan, M. U. Siddiqi, Transform domain characterization of abelian codes, IEEE Trans. Inform. Theory 38(6) (1992) 1817–1821.
17. [17] G. Séguin, A class of 1-generator quasi–cyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1745–1753.
18. [18] S. K. Wasan, Quasi abelian codes, Publ. Inst. Math. 21(35) (1977) 201–206.