New Classes of $p$-ary Few Weights Codes

TL;DR

This paper introduces new classes of three-weight and two-weight codes over chain rings, generalizing previous work, with detailed weight distributions, optimality conditions, and applications to secret sharing.

Contribution

It constructs and analyzes new classes of homogeneous weight codes over chain rings, extending prior results and exploring their optimality and minimality properties.

Findings

Homogeneous weight distributions are explicitly computed.

Conditions for optimal Gray images are established.

Codes are shown to be minimal, suitable for secret sharing.

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.