Few-weight codes over $\Bbb F_p+u\Bbb F_p$ associated with down sets and their distance optimal Gray image

TL;DR

This paper constructs multiple classes of few-weight codes over a specific ring linked to down sets, computes their weight distributions, and derives distance optimal codes over finite fields, including some that meet the Griesmer bound.

Contribution

It introduces new classes of few-weight codes over the ring $\mathbb{F}_p + u\mathbb{F}_p$, analyzes their weight distributions, and constructs distance optimal $p$-ary linear codes via Gray maps.

Findings

Computed Lee weight distributions for codes generated by single maximal elements.

Constructed $p$-ary distance optimal codes from the Gray images.

Identified codes that meet the Griesmer bound.

Abstract

Let $p$ be an odd prime number. In this paper, we construct $2(2p-3)$ classes of codes over the ring $R=\Bbb F_p+u\Bbb F_p,u^2=0$, which are associated with down sets. We compute the Lee weight distributions of the $2(2p-3)$ classes of codes when the down sets are generated by a single maximal element. Moreover, by using the Gray map of the linear codes over $R$, we find out $2(p-1)$ classes of $p$-ary distance optimal linear codes. Two of them meet the Griesmer bound.