Self-dual Hadamard bent sequences

TL;DR

This paper investigates self-dual Hadamard bent sequences, exploring their properties, generation methods, and automorphism groups, and conjectures their existence for even perfect square lengths.

Contribution

It introduces the study of self-dual Hadamard bent sequences, compares generation methods, and defines the strong automorphism group with an efficient computation algorithm.

Findings

Many examples from regular and Bush-type Hadamard matrices

Conjecture that self-dual bent sequences exist for all even perfect squares

Development of an efficient algorithm to compute the automorphism group

Abstract

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Sol\'e et al, 2021). We study the self dual class in length at most $196.$ We use three competing methods of generation: Exhaustion, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices provide many examples. We conjecture that if $v$ is an even perfect square, a self-dual bent sequence of length $v$ always exist. We introduce the strong automorphism group of Hadamard matrices, which acts on their associated self-dual bent sequences. We give an efficient algorithm to compute that group.