Binary Linear Codes From Vectorial Boolean Functions and Their Weight Distribution

TL;DR

This paper constructs new binary linear codes from vectorial Boolean functions, analyzing their parameters and weight distributions, and improves upon existing constructions by Ding et al. with larger dimensions.

Contribution

It introduces new classes of binary linear codes derived from vectorial Boolean functions, providing explicit parameters and weight distributions, especially for perfect nonlinear and Gold functions.

Findings

Constructed several classes of six-weight linear codes including the all-one codeword.

Determined the weight distribution of subcodes when using perfect nonlinear or Gold functions.

Achieved larger code dimensions compared to previous generic constructions.

Abstract

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding \emph{et al.} recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contains the all-one codeword. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.'s generic construction.