Hadamard matrices related to a certain series of ternary self-dual codes

TL;DR

This paper explores the relationship between Hadamard matrices and a specific series of ternary self-dual codes, revealing their structural connections and the existence of multiple inequivalent Hadamard matrices within a notable extremal code.

Contribution

It establishes that certain ternary self-dual codes contain Hadamard matrices and are generated by their rows, also identifying multiple inequivalent Hadamard matrices in a key extremal code.

Findings

Codes contain Hadamard matrices of order 2(p+1) for primes p ≡ 5 mod 24

Codes are generated by the rows of these Hadamard matrices

The extremal code of length 60 contains at least two inequivalent Hadamard matrices

Abstract

In 2013, Nebe and Villar gave a series of ternary self-dual codes of length $2(p+1)$ for a prime $p$ congruent to $5$ modulo $8$. As a consequence, the third ternary extremal self-dual code of length $60$ was found. We show that the ternary self-dual code contains codewords which form a Hadamard matrix of order $2(p+1)$ when $p$ is congruent to $5$ modulo $24$. In addition, it is shown that the ternary self-dual code is generated by the rows of the Hadamard matrix. We also demonstrate that the third ternary extremal self-dual code of length $60$ contains at least two inequivalent Hadamard matrices.