On the Structure of Linear and Cyclic Codes over a Finite Chain Ring
Copyright policy )On the Structure of Linear and Cyclic Codes over a Finite Chain Ring
In [\textit{A. R. Calderbank} and \textit{N. J. A. Sloane}, Des. Codes Cryptography 6, 21-35 (1995; Zbl 0848.94020)], structure theorems are given for linear and cyclic codes over \(\mathbb{Z}_{p^a}\). Such codes received much attention after the discovery that important families of binary nonlinear codes can be obtained as images under a Gray mapping of \(\mathbb{Z}_4\)-linear codes [\textit{A. R. Hammons jun., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane} and \textit{P. Solé}, IEEE Trans. Inf. Theory 40, 301-319 (1994; Zbl 0811.94039)]. The present paper generalizes the results of Calderbank-Sloane (loc. cit.) to a finite chain ring. New proofs are given that avoid the nontrivial commutative algebra invoked in Calderbank-Sloane (loc. cit.). In a follow-up paper [\textit{G. H. Norton} and \textit{A. Sălăgean}, IEEE Trans. Inf. Theory 46, 1060-1067 (2000; Zbl 0963.94043)], the authors use the structure theorems from this paper to obtain results on the minimum Hamming distance of codes over a finite chain ring.
- University of Bristol United Kingdom
- Nottingham Trent University United Kingdom
Algebraic coding theory; cryptography (number-theoretic aspects), cyclic code, linear code, finite chain ring, Galois ring, Linear codes (general theory), minimum Hamming distance
Algebraic coding theory; cryptography (number-theoretic aspects), cyclic code, linear code, finite chain ring, Galois ring, Linear codes (general theory), minimum Hamming distance
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