On the number of nonequivalent propelinear extended perfect codes

TL;DR

This paper demonstrates that the number of nonequivalent propelinear extended perfect binary codes grows exponentially with length, establishing their abundance and properties such as transitivity and normalized representations.

Contribution

It proves the exponential growth of nonequivalent propelinear extended perfect codes and shows all transitive codes by Potapov are propelinear with normalized representations.

Findings

Exponential number of nonequivalent codes as length increases

All transitive extended perfect codes by Potapov are propelinear

Codes have small rank, one more than extended Hamming code

Abstract

The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation.