Remark on a result of Constantine

TL;DR

This paper generalizes Constantine's result by constructing specific codes based on Hadamard matrices, extending the known cases from prime powers to more general orders.

Contribution

It introduces a new construction of codes of length 4n with specified size and minimum distance for orders related to Hadamard matrices, broadening previous results.

Findings

Codes of length 4n with 8n+8 codewords constructed

Minimum distance of these codes is 2n-2

Generalizes Constantine's prime power case

Abstract

In this short note we construct codes of length $4n$ with $8n+8$ codewords and minimum distance $2n-2$ whenever $4n+4$ is the order of a Hadamard matrix. This generalises work of Constantine who obtained a similar result in the special case that $n$ is a prime power.