Multi-dimensional Constacyclic Codes of Arbitrary Length over Finite Fields

TL;DR

This paper characterizes the algebraic structure of multi-dimensional constacyclic codes over finite fields, focusing on three-dimensional cases of arbitrary length and conditions for self-duality.

Contribution

It extends the theory of constacyclic codes to three dimensions and arbitrary lengths, providing necessary and sufficient conditions for self-duality.

Findings

Characterization of three-dimensional constacyclic codes over finite fields.

Conditions for self-duality of these codes.

Generalization to arbitrary length codes.

Abstract

Multi-dimensional cyclic code is a natural generalization of cyclic code. In an earlier paper we explored two-dimensional constacyclic codes over finite fields. Following the same technique, here we characterize the algebraic structure of multi-dimensional constacyclic codes, in particular three-dimensional $(\alpha,\beta,\gamma)$- constacyclic codes of arbitrary length $s\ell k$ and their duals over a finite field $\mathbb{F}_q$, where $\alpha,\beta,\gamma$ are non zero elements of $\mathbb{F}_q$. We give necessary and sufficient conditions for a three-dimensional $(\alpha,\beta,\gamma)$- constacyclic code to be self-dual.