Two Classes of Constacyclic Codes with Variable Parameters

TL;DR

This paper introduces two new classes of constacyclic codes constructed from cyclic codes, demonstrating their optimality and analyzing their parameters, which advances both theoretical understanding and practical applications of these codes.

Contribution

The paper presents novel constructions of two classes of constacyclic codes based on cyclic codes, expanding the family and analyzing their parameters.

Findings

Both classes contain optimal linear codes.

Parameters of the codes are thoroughly analyzed.

Open problems related to these codes are proposed.

Abstract

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed and some open problems are presented in this paper.