On repeated-root multivariable codes over a finite chain ring

Abstract

In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r -dimensional cyclic version.

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.…”

“…Properties of multivariable codes over a finite chain ring depend on the structure of the ambient ring R. So, in [16] a complete account of codes was given when the polynomials t i (X i ) ∈ F q [X i ] have no repeated roots (the so-called semisimple or serial case). On the other hand, as a first approach to the repeated-root (or modular ) case, Canonical Generating Systems [19] were considered in [17]. Unfortunately, the description is not as satisfactory as in the semisimple case.…”

Codes over affine algebras with a finite commutative chain coefficient ring

“…As an application of [20,Theorem 4.3] to affine algebras, we have the following result, which generalizes [1, Theorem 1] and also [16,Theorem 2]. Theorem 1.…”

Group algebras and coding theory

“…Codes over rings developed even more in the beginning of the 21st century that they deserved a CIMPA Summer School in 2008 [71]. Further works can be found in [18], [42], [47]. A small survey on the subject is [33].…”