Group LCD and Group Reversible LCD Codes

TL;DR

This paper introduces a novel method for constructing LCD codes using group rings, ensuring the codes are also group codes, and extends the construction to reversible LCD codes with examples including optimal codes.

Contribution

The paper presents a new construction technique for LCD and reversible LCD codes via group rings, expanding the class of such codes with explicit examples.

Findings

Constructed numerous binary group LCD codes, some optimal.

Developed conditions for non-trivial group LCD code construction.

Extended the method to produce reversible LCD codes.

Abstract

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element $v,$ one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element $v,$ one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.