Vanishing ideals of binary Hamming spheres

TL;DR

This paper studies the algebraic structure of Boolean functions that vanish on Hamming spheres, providing efficient computation methods and connections to the binary Möbius transform, with applications to coding theory.

Contribution

It introduces a periodicity property of the sANF vector, enabling efficient computation and explicit formulas, and links these to the binary Möbius transform, with applications to code distance bounds.

Findings

sANF vector is periodic, allowing efficient computation

Explicit formulas for specific cases of vanishing Boolean functions

A polynomial evaluation method to bound minimum distance of codes

Abstract

We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{\"o}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weight distribution of binary linear codes.