Several classes of linear codes with few weights derived from Weil sums

TL;DR

This work addresses constructing $q$-ary linear codes with few weights by defining the code length via the intersection, difference, and union of two trace-defined sets. It relies on Weil sums and Gauss sums to completely determine parameters and weight distributions for codes from $D_1$, $D_2$, and $D_3$, under the condition that $m/\gcd(m,e)$ is even. The authors establish complete weight distributions for $t$-weight codes with $t\in\{3,4,5,6\}$ and obtain several optimal codes meeting the Griesmer bound, with extensive case analyses and corroborating MAGMA experiments. The results provide a flexible framework for generating many (almost) optimal codes with potential applications in secret sharing, authentication, association schemes, and strongly regular graphs.

Abstract

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with the defining sets given by the intersection, difference and union of two certain sets, where $t=3,4,5,6$ and $q$ is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.

Theorems & Definitions (50)

• Lemma 1
• Lemma 2
• Lemma 3
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• Lemma 7
• Lemma 8
• Theorem 1
• Example 1