2024/12/21 by José Ándrés Armario, Armario, José Andrés, Ronan Egan +5 · 1 citation
Engineering · #05B20 #94A60 #94B25 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2412.16579
openalex publication_date 2024/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An n× n complex matrix M with entries in the k^\textrmth roots of unity which satisfies MM∗ = nIn is called a Butson Hadamard matrix. While a matrix with entries in the k^\textrmth 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^\textrmth roots of unity and all entries of 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\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.