A generalisation of bent vectors for Butson Hadamard matrices
José Andrés Armario, Ronan Egan, Hadi Kharaghani, Padraig Ó Catháin
TL;DR
The paper generalizes bent vectors to Butson Hadamard matrices by defining $H$-bent vectors and their self-dual and conjugate self-dual variants, and then applying algebraic number theory to derive order restrictions and nonexistence results. It develops explicit constructions via tensor products and Bush-type matrices to produce numerous bent vectors, including conjugate self-dual cases, and explores special matrix equations that yield additional families of bent vectors. A key application is obtained by analyzing the covering radius of BH-codes, linking bent-vector structure to bounds on nonlinearity for generalized Reed–Muller-type codes. Overall, the work provides both theoretical obstructions and constructive tools for bent vectors in the wider BH$(n,k)$ framework, with implications for code performance and design theory.
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 $M{\bf x} = λ{\bf y}$ 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 ${\bf y} = μ{\bf x}$ and conjugate self-dual if ${\bf y} = μ\overline{\bf x}$ for some $μ$ 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.