A generalisation of bent vectors for Butson Hadamard 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.