The homogeneous weight for $R_k$, related Gray map and new binary quasicyclic codes

TL;DR

This paper characterizes the homogeneous weight for the ring family R_k, constructs a Gray map, and uses these to generate many new optimal binary quasicyclic codes with desirable properties.

Contribution

It introduces the homogeneous weight for R_k, develops a Gray map based on Reed-Muller codes, and constructs numerous new optimal binary quasicyclic codes.

Findings

Identified the homogeneous weight for R_k rings.

Constructed a Gray map linking codes over R_k to binary codes.

Discovered many new optimal binary quasicyclic codes of indices 8, 16, and 24.

Abstract

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family $R_k$, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over $R_k$ under this Gray map. We then discuss quasitwisted codes over $R_k$ and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasicyclic. In particular, we find a substantial number of optimal binary codes that are quasicyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasicyclic codes kept by Chen.