Propelinear 1-perfect codes from quadratic functions

TL;DR

This paper demonstrates that perfect codes constructed using quadratic functions from a linear base are propelinear and transitive, significantly increasing the known number of such codes compared to previous bounds.

Contribution

It introduces a method to generate a vast number of propelinear 1-perfect codes from quadratic functions, expanding the understanding of their structure and abundance.

Findings

At least exp(cN^2) propelinear 1-perfect codes of length N exist.

The number of transitive codes is bounded above by exp(C(N ln N)^2).

Quadratic functions enable the construction of highly symmetric perfect codes.

Abstract

Perfect codes obtained by the Vasil'ev--Sch\"onheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least $\exp(cN^2)$ propelinear $1$-perfect codes of length $N$ over an arbitrary finite field, while an upper bound on the number of transitive codes is $\exp(C(N\ln N)^2)$. Keywords: perfect code, propelinear code, transitive code, automorphism group, Boolean function.