Constructing cocyclic Hadamard matrices of order 4p

TL;DR

This paper develops a classification method for cocyclic Hadamard matrices of order 4p, where p is prime, revealing structural patterns and providing classifications for primes up to 13.

Contribution

It introduces a new classification algorithm for CHMs of order 4p, proves refined structural results, and classifies all such matrices for primes p ≤ 13.

Findings

Every CHM of order 4p with p ≡ 1 mod 4 is equivalent to a matrix with one of five block structures.

For p ≡ 3 mod 4, CHMs are equivalent to Williamson or Ito type matrices.

Complete classification achieved for primes p ≤ 13.

Abstract

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.