Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions

TL;DR

This paper explores the dual and Hull codes within two generic constructions, linking their properties to Walsh transforms of cryptographic functions, and introduces methods to explicitly compute and analyze these codes.

Contribution

It is the first to study dual codes in this framework, proposing a Gram-Schmidt process and analyzing the existence of defining sets for dual code construction.

Findings

Explicit Gram-Schmidt process for dual codes

Necessary Walsh transform condition for codeword dual membership

Novel description of Hull codes in the generic construction framework

Abstract

We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. This is the first work in which we study the dual codes in the framework of the two generic constructions; in particular, we propose a Gram-Schmidt (complexity of $\mathcal{O}(n^3)$) process to compute them explicitly. The originality of this contribution is in the study of the existence or not of defining sets $D'$, which can be used as ingredients to construct the dual code $\mathcal{C}'$ for a given code $\mathcal{C}$ in the context of the second generic construction. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of $\mathcal{C}$ to belong in the dual. This achievement was done in general and when the involved functions are weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals.