A Circulant Approach to Skew-Constacyclic Codes

TL;DR

This paper introduces a circulant matrix framework for skew-constacyclic codes, enabling new insights into their structure and duality properties within skew-polynomial rings.

Contribution

It develops a circulant-based approach to analyze skew-constacyclic codes, including properties of these circulants and their relation to code duality, especially for two-sided polynomials.

Findings

Circulant matrices effectively represent skew-polynomial structures.

The transpose of a circulant remains a circulant, aiding dual code analysis.

The dual of a skew-constacyclic code is also a constacyclic code.

Abstract

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x;\theta] modulo the left ideal generated by a polynomial of the type x^n-a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This recovers the well-known result that the dual of a skew-constacyclic code is a constacyclic code again. Special attention is paid to the case where x^n-a is two-sided.