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

Abstract

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 $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrix is a G-invariant $$|G|\times |G|$$ matrix H whose entries are complex hth roots of unity such that $$HH^*=|G|I_{|G|}$$. By $$\nu _p(x)$$ we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrices under the following conditions. The purpose of this paper is to further study the conditions on G and h so that a $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrix exists. We will focus on $${{\,\mathrm{\mathrm{BH}}\,}}({\mathbb {Z}}_n,h)$$ and $${{\,\mathrm{\mathrm{BH}}\,}}(G,2p^b)$$ matrices, where p is an odd prime. Our results describe various relation among |G|, $$\gcd (|G|,h)$$ and $${{\,\mathrm{\mathrm{lcm}}\,}}(|G|,h)$$. Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of $${{\,\mathrm{\mathrm{BH}}\,}}(\mathbb {Z}_n,h)$$ matrices in which $$1\le n,h\le 100$$. Finally, we show that $${{\,\mathrm{\mathrm{BH}}\,}}(G,h)$$ matrices can be used to construct a new family of perfect polyphase arrays.