Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Abstract

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p a , m) and generating sets for its ideals are considered. It is shown that these generating sets are strong Groebner bases. A method for finding such sets in the case that a = 2 is also given. The Hamming distance of certain constacyclic codes of length ηp s and 2ηp s over Fpm is computed. A method, which determines the Hamming distance of the constacyclic codes of length ηp s and 2ηp s over GR(p a , m), where (η, p) = 1, is described. In particular, the Hamming distance of all cyclic codes of length p s over GR(p 2 , m) and all negacyclic codes of length 2p s over Fpm is determined explicitly.

Multivariable codes in principal ideal polynomial quotient rings with applications to additive modular bivariate codes overF4

“…As it was shown in [4, Theorem 2], we have the following characterization (see also [12,Theorem 5.2], [22,Theorem 3.2], [17,Theorem 1]). Theorem 1.…”

Do non-free LCD codes over finite commutative Frobenius rings exist?

“…From [19,Theorem 5.2. ], the quotient ring R [X ]/〈 X n − γ 〉 is a principal ideal ring, if either R is a field, or X n − π(γ) is free-square.…”

On polycyclic codes over a finite chain ring

“…Note that the quotient ring S[X]/ X n − Ψ( a ) is a principal ideal ring if either S is a field or X n − π(Ψ( a )) is square free [8,Theorem 5.2.]. From now on we will consider a-cyclic code over S whose period is relatively prime to p and hence X n − π(Ψ( a )) is square free.…”

“…Throughout this work, S is a commutative ring with identity, and Aut(S) is the group of ring automorphisms of S. The action of Aut(S) on S n leads to the concept of Galois-invariance for any S-linear code of length n (see [12]). In [8] the Hamming Figure 1. Cyclicity of codes weight of some polycyclic codes over Galois rings was established.…”