Linear Codes over $\mathbb{F}_{q}[x]/(x^2)$ and $GR(p^2,m)$ Reaching the Griesmer Bound

TL;DR

This paper constructs new linear codes over specific finite rings that reach the Griesmer bound, and derives related codes over finite fields with optimal properties, including two-weight codes.

Contribution

It introduces two series of codes over rings that achieve the Griesmer bound and derives finite field codes with optimal parameters from them.

Findings

Codes over rings reach the Griesmer bound.

Derived codes over finite fields are linear and sometimes reach the Griesmer bound.

Many codes have two Hamming weights.

Abstract

We construct two series of linear codes over finite ring $\mathbb{F}_{q}[x]/(x^2)$ and Galois ring $GR(p^2,m)$ respectively reaching the Griesmer bound. They derive two series of codes over finite field $\mathbb{F}_{q}$ by Gray map. The first series of codes over $\mathbb{F}_{q}$ derived from $\mathbb{F}_{q}[x]/(x^2)$ are linear and also reach the Griesmer bound in some cases. Many of linear codes over finite field we constructed have two Hamming (non-zero) weights.