Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays

• Main Author: Duc, Tai Do
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2020
• Subjects: Science::Mathematics; Group Invariant Butson Hadamard Matrices; Perfect Arrays
• Online Access: https://hdl.handle.net/10356/143067
• Institution: Nanyang Technological University

Let G be a finite abelian group and let exp (G) denote the least common multiple of the orders of all elements of G. A BH(G,h) matrix is a G-invariant | G| × | G| matrix H whose entries are complex hth roots of unity such that HH∗= | G| I|G|. By νp(x) we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of BH(G,h) matrices under the following conditions.(i)νp(h) ≥ ⌈ νp(exp (G)) / 2 ⌉ for any prime divisor p of |G|, and(ii)ν2(h) ≥ 2 if ν2(| G|) is odd and G has a direct factor Z2. The purpose of this paper is to further study the conditions on G and h so that a BH(G,h) matrix exists. We will focus on BH(Zn,h) and BH(G,2pb) matrices, where p is an odd prime. Our results describe various relation among |G|, gcd (| G| , h) and lcm(|G|,h). Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of BH(Zn,h) matrices in which 1 ≤ n, h≤ 100. Finally, we show that BH(G,h) matrices can be used to construct a new family of perfect polyphase arrays.