2018/07/11 by Ronan Egan, Padraig Ó Catháin, Egan, Ronan +3
Engineering · Mathematics · #05B20 #05B30 #05C50 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1807.04238
openalex publication_date 2018/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Butson Hadamard matrix H has entries in the kth roots of unity, and satisfies the matrix equation HH∗ = nIn. We write BH(n, k) for the set of such matrices. A complete morphism of Butson matrices is a map BH(n, k) → BH(m, ℓ). In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer t, we construct a real Hadamard matrix Ht of order nt = 2^2t-1-1 such that the minimal polynomial of \frac1√ntHt is the cyclotomic polynomial Φ2t+1(x). Such matrices yield new examples of complete morphisms BH(n, 2t) → BH(2^2t-1-1n, 2) , for each t ≥ 2, generalising a well-known result of Turyn.