New Classes of $p$-ary Few Weights Codes

TL;DR

This paper constructs new classes of three-weight and two-weight codes over chain rings, analyzes their weight distributions, optimality, and minimality, with potential applications in secret sharing schemes.

Contribution

It introduces new classes of trace codes over chain rings with specific weight properties, generalizing previous work and analyzing their optimality and minimality.

Findings

Homogeneous weight distributions are computed using exponential sums.

Conditions for optimal Gray images are established based on the Griesmer bound.

Codewords are shown to be minimal, supporting secret sharing applications.

Abstract

In this paper, several classes of three-weight codes and two-weight codes for the homogeneous metric over the chain ring $R=\mathbb{F}_p+u\mathbb{F}_p+\cdots +u^{k-1}\mathbb{F}_{p},$ with $u^k=0,$ are constructed, which generalises \cite{SL}, the special case of $p=k=2.$ These codes are defined as trace codes. In some cases of their defining sets, they are abelian. Their homogeneous weight distributions are computed by using exponential sums. In particular, in the two-weight case, we give some conditions of optimality of their Gray images by using the Griesmer bound. Their dual homogeneous distance is also given. The codewords of these codes are shown to be minimal for inclusion of supports, a fact favorable to an application to secret sharing schemes.