Linear codes with few weights over $\mathbb{F}_{p}+u\mathbb{F}_{p}$

TL;DR

This paper constructs linear codes over a specific ring extension of finite fields using trace functions and determines their weight distributions through symplectic-weight analysis of Gray images.

Contribution

It introduces a method to construct linear codes over _{p}+u_{p} using trace functions and analyzes their weight distributions.

Findings

Derived explicit weight distributions for the constructed codes.

Established connections between symplectic weights and Gray images.

Provided new classes of codes with few weights.

Abstract

For any positive integer $m$ and an odd prime $p$; let $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $q=p^{m}$, be a ring extension of the ring $\mathbb{F}_{p}+u\mathbb{F}_{p}.$ In this paper, we construct linear codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$ by using trace function defined on $\mathbb{F}_{q}+u\mathbb{F}_{q}$ and determine their Hamming weight distributions by employing symplectic-weight distributions of their Gray images.