On abelian and cyclic group codes

2022/09/28 by Angelo Marotta, Marotta, Angelo
Computer Science · Engineering · Mathematics · #Abelian group #Algorithm #Automorphism #Block code #Code (set theory) #Coding theory and cryptography #Combinatorics #Computer science #Cooperative Communication and Network Coding #Cyclic code #Cyclic group #Discrete mathematics #Equivalence (formal languages) #FOS: Computer and information sciences #FOS: Mathematics #Group (periodic table) #Group Theory (math.GR) #Group code #Hamming code #Hamming distance #Information Theory (cs.IT) #Linear code #Mathematics #Permutation (music) #Physics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2209.14213

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2022/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of s copies of the repetition code of length t, for some suitable positive integers s and t. Moreover, we provide a complete characterisation of permutation automorphisms of the linear code C=\bigoplus_i=1^sRep_t(\mathbbF_q) and we establish that such a code is an abelian group code, for every pair of integers s,t≥1. Finally, in a similar fashion as for abelian group codes, we give an equivalent characterisation of cyclic group codes.

Related

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