Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31

TL;DR

This paper classifies specific Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31, and explores their implications for constructing new self-dual codes with previously unknown properties.

Contribution

It provides a classification of Hadamard matrices with particular automorphisms and constructs new self-dual codes, expanding the known catalog of such codes.

Findings

New Hadamard matrices of order 60 and 64 identified

New ternary near-extremal self-dual codes discovered

New binary near-extremal doubly even self-dual codes with unknown weight enumerators

Abstract

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$ are computed, as well as the binary doubly even self-dual codes of length $120$ with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.