Linear codes with a few weights from inhomogeneous quadratic functions

TL;DR

This paper constructs linear codes with few weights using inhomogeneous quadratic functions over finite fields, expanding on existing codes and determining their weight distributions for applications in cryptography and combinatorics.

Contribution

It introduces a new class of linear codes derived from inhomogeneous quadratic functions and provides their weight distributions, including some previously known codes as special cases.

Findings

Constructed new linear codes with few weights from inhomogeneous quadratic functions.

Determined the weight distributions of these codes.

Included some known codes as special cases.

Abstract

Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes with a few weights are constructed from inhomogeneous quadratic functions over the finite field $\gf(p)$, where $p$ is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.