A generalisation of bent vectors for Butson Hadamard matrices

TL;DR

This paper generalizes the concept of bent vectors for Butson Hadamard matrices, exploring their properties, existence conditions, and applications to coding theory using algebraic number theory and tensor constructions.

Contribution

It introduces a generalization of bent vectors for Butson Hadamard matrices, providing order conditions, non-existence results, explicit examples, and applications to coding theory.

Findings

Order conditions for self-dual and conjugate self-dual bent vectors

Non-existence results for certain bent vectors

Explicit constructions using tensor and Bush-type matrices

Abstract

An $n\times n$ complex matrix $M$ with entries in the $k^{\textrm{th}}$ roots of unity which satisfies $MM^{\ast} = nI_{n}$ is called a Butson Hadamard matrix. While a matrix with entries in the $k^{\textrm{th}}$ roots typically does not have an eigenvector with entries in the same set, such vectors and their generalisations turn out to have multiple applications. A bent vector for $M$ satisfies $M{\bf x} = \lambda {\bf y}$ where ${\bf x}$ has entries in the $k^{\textrm{th}}$ roots of unity and all entries of $\textbf{y}$ are complex numbers of norm $1$. Such a bent vector ${\bf x}$ is self-dual if ${\bf y} = \mu{\bf x}$ and conjugate self-dual if ${\bf y} = \mu\overline{\bf x}$ for some $\mu$ of norm $1$. Using techniques from algebraic number theory, we prove some order conditions and non-existence results for self-dual and conjugate self-dual bent vectors; using tensor constructions and Bush-type matrices we give explicit examples. We conclude with an application to the covering radius of certain non-linear codes generalising the Reed Muller codes.