Transitive nonpropelinear perfect codes

TL;DR

This paper investigates the structure of transitive perfect codes, identifying a unique nonpropelinear code among length 15 and establishing the existence of multiple such codes for all larger admissible lengths.

Contribution

It proves the existence of transitive nonpropelinear perfect codes for all admissible lengths n ≥ 15 and at least five pairwise nonequivalent codes for n ≥ 255.

Findings

Unique nonpropelinear code among length 15 codes.

Existence of transitive nonpropelinear perfect codes for all n ≥ 15.

At least five such codes for all n ≥ 255.

Abstract

A code is called transitive if its automorphism group (the isometry group) of the code acts transitively on its codewords. If there is a subgroup of the automorphism group acting regularly on the code, the code is called propelinear. Using Magma software package we establish that among 201 equivalence classes of transitive perfect codes of length 15 from \cite{ost} there is a unique nonpropelinear code. We solve the existence problem for transitive nonpropelinear perfect codes for any admissible length $n$, $n\geq 15$. Moreover we prove that there are at least 5 pairwise nonequivalent such codes for any admissible length $n$, $n\geq 255$.