Weight distributions of two classes of linear codes

TL;DR

This paper constructs two classes of linear codes over finite fields, determines their weight distributions using advanced mathematical tools, and identifies some optimal codes with applications in secret sharing schemes.

Contribution

It introduces new classes of at most six-weight linear codes with explicitly determined weight distributions, some of which are optimal or nearly optimal.

Findings

Some codes are almost optimal according to Griesmer bound.

The codes have applications in secret sharing schemes.

Explicit weight distributions are derived using Gaussian periods and Weil sums.

Abstract

In this paper, based on the theory of defining sets, two classes of at most six-weight linear codes over $\mathbb{F}_p$ are constructed. The weight distributions of the linear codes are determined by means of Gaussian period and Weil sums. In some case, there is an almost optimal code with respect to Griesmer bound, which is also an optimal one according to the online code table. The linear codes can also be employed to get secret sharing schemes.