Uniqueness of certain completely regular Hadamard codes

TL;DR

This paper classifies and proves the uniqueness of certain binary completely regular Hadamard codes of specific lengths and distances, establishing their automorphism groups and transitivity properties.

Contribution

It provides a complete classification and uniqueness proof for these codes, linking them to Mathieu groups and their transitivity.

Findings

Codes are unique up to equivalence for given parameters.

Automorphism groups are isomorphic to Mathieu groups.

Codes are necessarily completely transitive.

Abstract

We classify binary completely regular codes of length $m$ with minimum distance $\delta$ for $(m,\delta)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard codes. We prove that the automorphism groups of these Hadamard codes, modulo the kernel of a particular action, are isomorphic to certain Mathieu groups, from which we prove that completely regular codes with these parameters are necessarily completely transitive.