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

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.