New few weight codes from trace codes over a local Ring

TL;DR

This paper constructs new few-weight linear codes over a specific local ring using trace functions, evaluates their weight distributions, and explores their optimality and applications in secret sharing schemes.

Contribution

It introduces novel few-weight codes over a local ring via trace functions, analyzes their weight distributions, and demonstrates their optimality and utility in secret sharing.

Findings

Two-weight codes are optimal by Griesmer bound.

New three-weight codes are constructed under specific conditions.

Codes have applications in secret sharing schemes.

Abstract

In this paper, new few weights linear codes over the local ring $R=\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p,$ with $u^2=v^2=0, uv=vu,$ are constructed by using the trace function defined over an extension ring of degree $m.$ %In fact, These codes are punctured from the linear code is defined in \cite{SWLP} up to coordinate permutations. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gaussian sums over finite fields. Two different defining sets are explored. Using a linear Gray map from $R$ to $\mathbb{F}_p^4,$ we obtain several families of new $p$-ary codes from trace codes of dimension $4m$. For the first defining set: when $m$ is even, or $m$ is odd and $p\equiv3 ~({\rm mod} ~4),$ we obtain a new family of two-weight codes, which are shown to be optimal by the application of the Griesmer bound; when $m$ is even and under some special conditions, we obtain two new classes of three-weight codes. For the second defining set: we obtain a new class of two-weight codes and prove that it meets the Griesmer bound. In addition, we give the minimum distance of the dual code. Finally, applications of the $p$-ary image codes in secret sharing schemes are presented.