SavedAdd your journal

Journals / Turkish Journal of Mathematics / 2021 / Cilt: 45 - Sayı: 1

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

İpek TUVAY
Fatma ALTUNBULAK AKSU
Pages
445–455
Publication Date
2022-11-01
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let G be a finite abelian group.
📊
Key Findings & Data: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.
💡
Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:2014 Verileri
⚙️Variables:Domain Parameters

Abstract

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.

🔖

Cite This Article

TUVAY, İ., AKSU, F. A. (2022). On the number of non-G-equivalent minimal abelian codes. Turkish Journal of Mathematics, Cilt: 45 - Sayı: 1, 445–455. https://ajindex.com/articles/608681

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (16)

  1. Assuena S, Polcino Milies, C. Good codes from metacyclic groups. Contemporary Mathematics-American Mathematical Society 2019
  2. 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
  4. doi: 10.1007/BF01072842 [4] Bernal J, del Río Á, Simón JJ. An intrinsical description of group codes. Codes and Cryptography 2009
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. (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
  15. 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
  16. doi: 10.1007/s10623-0