Automorphisms of Kimura Hadamard Matrices

TL;DR

This paper analyzes the automorphism groups of Kimura Hadamard matrices derived from dihedral groups, revealing their structure, additional automorphisms, and counterexamples to existing conjectures.

Contribution

It characterizes the automorphism groups of KHMs from dihedral groups, identifies new automorphisms, and challenges previous conjectures about their structure.

Findings

Automorphism groups always contain a subgroup isomorphic to D_{2k}×Q_8 or C_2×D_{2k}×Q_8.

Additional automorphisms arise from the holomorph of the dihedral group under certain conditions.

Counterexamples to a conjecture of Ó Cathaín are provided, indicating no further automorphisms beyond those identified.

Abstract

We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms, and show that the automorphism group of a KHM always has a subgroup isomorphic to $D_{2k}\times Q_8$, or $C_2\times D_{2k}\times Q_8$ if it is $y$-invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda--Yamada construction, reveals counterexamples to a conjecture of \'O Catha\'{\i}n and suggests that no further automorphisms occur beyond those predicted by our framework.