On the number of non-G-equivalent minimal abelian codes

Let G be a finite abelian group. Ferraz, Guerreiro, and Polcino Milies (2014) proved that the number of G-equivalence classes of minimal abelian codes is equal to the number of G-isomorphism classes of subgroups for which corresponding quotients are cyclic. In this article, we prove that the notion of G-isomorphism is equivalent to the notion of isomorphism on the set of all subgroups H of G with the property that G/H is cyclic. As an application, we calculate the number of non-G-equivalent minimal abelian codes for some specific family of abelian groups. We also prove that the number of non-G-equivalent minimal abelian codes is equal to the number of divisors of the exponent of G if and only if for each prime p dividing the order of G, the Sylow p-subgroups of G are homocyclic.

PDF

___

[1] Assuena S, Polcino Milies, C. Good codes from metacyclic groups. Contemporary Mathematics-American Mathematical Society 2019; 727: 39-47. doi: 10.1090/conm/727
[2] Berman SD. Semisimple cyclic and abelian codes II. Cybernetics and Systems Analysis 1967; (3): 17-23. doi: 10.1007/BF01119999
[3] Berman SD. On the theory of group codes. Cybernetics 1967; 3: 25-31. doi: 10.1007/BF01072842
[4] Bernal J, del Río Á, Simón JJ. An intrinsical description of group codes. Codes and Cryptography 2009; 51: 289-300. doi: 10.1007/s10623-008-9261-z
[5] Borello M, Willems W. Group codes over fields are asymptotically good. Finite Fields and Their Applications 2020; 68: 101738. doi: 10.1016/j.ffa.2020.101738
[6] Chalom G, Ferraz RA, Guerreiro M. Minimal ideals in finite abelian group algebras and coding theory. São Paulo Journal of Mathematical Sciences 2016; 10: 321-340. doi: 10.1007/s40863-015-0037-x
[7] Ferraz RA, Guerreiro M, Polcino Milies C. G-equivalence in group algebras and minimal abelian codes. IEEE Transactions on Information Theory 2014; 60 (1): 252-260. doi: 10.1109/TIT.2013.2284211
[8] García Pillado C, González S, Markoc V, Markova O, Martínez C. Group codes of dimension 2 and 3 are abelian. Finite Fields and Their Applications 2019; 55: 167-176. doi: 10.1016/j.ffa.2018.09.009
[9] Jitman S, Ling S, Liu H, Xie X. Abelian Codes in Principal Ideal Group Algebras. IEEE Transactions on Information Theory 2013; 59(5): 3046-3058. doi: 10.1109/TIT.2012.2236383
[10] Kelarev AV, Solé P. Error-correcting codes as ideals in group rings. Contemporary Mathematics-American Mathematical Society 2001; 273: 11-18. doi: 10.1090/conm/273/04419
[11] MacWilliams FJ. Binary codes which are ideals in the group algebra of an abelian group. The Bell System Technical Journal 1970; 49 (6): 987-1011. doi: 10.1002/j.1538-7305.1970.tb01812.x
[12] Miller RL. Minimal codes in abelian group algebras. Journal of Combinatorial Theory Series A 1979; 26 (2): 166-178. doi:10.1016/0097-3165(79)90066-9
[13] Nomura T, Fukuda A. Linear recurring planes and two-dimensional linear recurring arrays. Electronics and Communications in Japan 1971; (54-A): 23-30.
[14] Nomura T, Miyakawa H, Imai H, Fukuda A. A theory of two-dimensional linear recurring arrays. IEEE Transactions on Information Theory 1972; 18 (6): 775-785. doi: 10.1109/TIT.1972.1054914
[15] Olteanu G, Van Gelder I. Construction of minimal non-abelian left group codes. Designs, Codes and Cryptography 2015; 75: 359-373. doi: 10.1007/s10623-014-9922-z
[16] Polcino Milies C, de Melo FD. On cyclic and abelian codes. IEEE Transactions on Information Theory 2013; 59(11): 7314-7319. doi: 10.1109/TIT.2013.2275111
[17] Polcino Milies C. Group algebras and coding theory: a short survey. Revista Integración 2019; 37: 153-166. doi: 10.18273/revint.v37n1-2019008