Constructions of Butson Hadamard matrices invariant under Abelian p-groups

Abstract

Let a and h be positive integers and let p be a prime. Let q1,…,qt be the distinct prime divisors of h and write Q(h)={∑i=1tciqi:ci∈Z,ci≥0}. We provide constructions of group invariant Butson Hadamard matrices BH(G,h) in the following cases. 1. G=(Zp)2a and at least one of the following conditions is satisfied. • pa∈Q(h), • pa+2∈Q(h) and h is even, • pa+1=(q1−1)(q2−1) where q1 and q2 are distinct prime divisors of h. 2. G=Zpa×Zpa and p−1,p∈Q(h). 3. G=(Zp2)a and pb∈Q(h) for some divisor b of a with 1≤b<a. 4. G=P×Zpa where P is any abelian group of order pa and p∈Q(h).