Two-Weight and a Few Weights Trace Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

TL;DR

This paper constructs infinite families of trace codes over the ring _q + u_q with few Lee-weights, including two-weight and multi-weight codes, using Gauss sums and analyzing their weight distributions.

Contribution

It introduces new infinite families of trace codes over _q + u7f_q with controlled weight distributions, including two-weight codes meeting the Griesmer bound.

Findings

Constructed codes have few Lee-weights, including two-weight and at most five-weight codes.

Codes meet the Griesmer bound when gd(e,m)=1.

New infinite families of codes are obtained for specific gcd conditions.

Abstract

Let $p$ be a prime number, $q=p^s$ for a positive integer $s$. For any positive divisor $e$ of $q-1$, we construct an infinite family codes of size $q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the ring $R=\mathbb{F}_q + u\mathbb{F}_q$, $u^2 = 0$. Using Gauss sums, their Lee weight distributions are provided. When $\gcd(e,m)=1$, we obtain an infinite family of two-weight codes over the finite field $\mathbb{F}_q$ which meet the Griesmer bound. Moreover, when $\gcd(e,m)=2, 3$ or $4$ we construct new infinite family codes with at most five-weight.