A nonexistence criterion and new constructions for Butson Hadamard matrices

TL;DR

The paper addresses the existence problem for Butson Hadamard matrices $BH(n,q)$ by developing a new necessary condition derived from positive definite functions on finite groups, encapsulated by a frequency map $g_H$ and its nonnegative Fourier transform $\widehat{g}$. This framework enables targeted nonexistence results for select $(n,q)$ and guides computational searches, demonstrated by the discovery of a BH$(18,14)$ and by producing new $2$-circulant constructions $BH(34,10)$, $BH(62,6)$, $BH(82,6)$, and $BH(146,6)$ via cyclotomic cosets and index-$k$ circulants. The paper also develops and applies a constructive approach for $2p$-order BH matrices through complementary pairs in index-$k$ circulant schemes, yielding several new examples and strengthening support for conjectures on $BH(2p,6)$ existence. While the condition proves nonexistence in many cases, the authors also illustrate its non-sufficiency with counterexamples, delineating the boundary between necessary criteria and constructive existence.

Abstract

Based on the concept of positive definite functions on finite groups, we present a new necessary condition for the existence of Butson Hadamard matrices $BH(n,q)$. We use this condition to prove some nonexistence results for a sequence of values of $(n,q)$, and also to facilitate a computer search and discover a matrix $BH(18,14)$. Furthermore, we use cyclotomic cosets to construct matrices $BH(34,10)$, $BH(62,6)$, $BH(82,6)$, and $BH(146,6)$ for the first time. These matrices have a $2$-circulant structure.

Theorems & Definitions (14)

• Conjecture 1
• Lemma 1
• proof
• Theorem 1
• Theorem 2: cf. M
• proof
• Theorem 3: Banica et al. BBS
• proof
• Theorem 4: cf. H
• proof