Binary linear codes with few weights from Boolean functions

TL;DR

This paper constructs binary linear codes with few weights using Boolean functions with specific properties, providing explicit parameters and potential applications in cryptography and combinatorics.

Contribution

It introduces new methods to build three- or four-weight binary linear codes from Boolean functions with limited Walsh transform values, including their parameters and duals.

Findings

Many classes of codes with explicit weight enumerators are obtained.

Some codes and their duals are optimal or nearly optimal.

The codes have potential applications in secret sharing, association schemes, and strongly regular graphs.

Abstract

Boolean functions have very nice applications in cryptography and coding theory, which have led to a lot of research focusing on their applications. The objective of this paper is to construct binary linear codes with few weights from the defining set, which is defined by some special Boolean functions and some additional restrictions. First, we provide two general constructions of binary linear codes with three or four weights from Boolean functions with at most three Walsh transform values and determine the parameters of their dual codes. Then many classes of binary linear codes with explicit weight enumerators are obtained. Some binary linear codes and their duals obtained are optimal or almost optimal. The binary linear codes obtained in this paper may have a special interest in secret sharing schemes, association schemes, strongly regular graphs.