Hadamard Structures with Associated Automorphisms

TL;DR

This paper introduces new Hadamard matrices and related combinatorial structures, including a large number of inequivalent matrices and designs, supporting the conjecture that all such matrices of order 36 are regular.

Contribution

It constructs a vast number of inequivalent Hadamard matrices and designs of order 36, and demonstrates their equivalence to regular matrices, advancing understanding of their structure.

Findings

Constructed 5202 inequivalent Hadamard matrices of order 36

Generated 180538 symmetric Hadamard designs with 35 points

All constructed matrices are equivalent to regular Hadamard matrices

Abstract

In this paper we present new Hadamard matrices and related combinatorial structures. In particular, it is constructed 5202 inequivalent Hadamard matrices of order 36 as well as 180538 Hadamard symmetric designs with 35 points in addition to those structures that admit an automorphism of order 3. Consequently, there are at least 272116 Hadamard 3-designs with 36 points and 70 lines. We found that all Hadamard matrices constructed here are equivalent to a regular Hadamard matrix. This fact contributes to the conjecture that Hadamard matrices of order 36, and possibly those of order $4m^2$, are regular.