Binary linear codes with few weights from two-to-one functions

TL;DR

This paper constructs binary linear codes with few weights using two-to-one functions over finite fields, analyzing their properties and showing some are optimal or have best known parameters.

Contribution

It introduces new classes of codes from two-to-one functions, including explicit weight distributions and optimality results.

Findings

Codes with one, three, four, and five weights are constructed.

Some codes achieve the sphere packing bound.

Several codes are optimal or have best known parameters.

Abstract

In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$ with $\gcd(t, n)=1$ and $\gcd\left(e, 2^n-1\right)=1$. Based on the study of the Walsh transforms of those functions or their related-ones, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. { Moreover, several examples show that some of our codes are optimal and some have the best known parameters.}