One–generator quasi–abelian codes revisited

Abstract

The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl’s online table are presented.

Keywords

• Group algebras,Quasi-abelian codes,Minimum distances,1-generator

References

1. [1] S. D. Berman, Semi–simple cyclic and abelian codes. II, Kibernetika 3(3) (1967) 21–30.
2. [2] S. D. Berman, On the theory of group codes, Kibernetika 3(1) (1967) 31–39.
3. [3] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265.
4. [4] C. Ding, D. R. Kohel, S. Ling, Split group codes, IEEE Trans. Inform. Theory 46(2) (2000) 485–495.
5. [5] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, Accessed on 2015-10-09.
6. [6] S. Jitman, Generator matrices for new quasi–abelian codes, Online available at https://sites.google.com/site/quasiabeliancodes, Accessed on 2015-10-09.
7. [7] S. Jitman, S. Ling, Quasi–abelian codes, Des. Codes Cryptogr. 74(3) (2015) 511–531.
8. [8] K. Lally, P. Fitzpatrick, Algebraic structure of quasicyclic codes, Discrete Appl. Math. 111(1–2) (2001) 157–175.

9. [9] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47(7) (2001) 2751–2760.
10. [10] S. Ling, P. Solé, Good self–dual quasi–cyclic codes exist, IEEE Trans. Inform. Theory 49(4) (2003) 1052–1053.
11. [11] S. Ling, P. Solé, On the algebraic structure of quasi–cyclic codes III: Generator theory, IEEE Trans. Inform. Theory 51(7) (2005) 2692–2700.
12. [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, Amsterdam, The Netherlands: North–Holland, 1977.
13. [13] J. Pei, X. Zhang, 1-generator quasi–cyclic codes, J. Syst. Sci. Complex. 20(4) (2007) 554–561.
14. [14] G. E. Seguin, A class of 1-generator quasi–cyclic codes, IEEE Trans. Inform. Theory 50(8) (2004) 1745–1753.
15. [15] S. K. Wasan, Quasi abelian codes, Pub. Inst. Math. 21(35) (1977) 201–206.