On the structure of non-full-rank perfect codes

TL;DR

This paper generalizes the structure and construction methods of non-full-rank perfect error-correcting codes from binary to q-ary cases, introducing new bounds and component-based assembly techniques.

Contribution

It extends the Krotov and Heden construction framework to q-ary codes and introduces a novel component-based approach for constructing perfect codes.

Findings

Every non-full-rank perfect code can be decomposed into well-defined components.

Components from different codes can be combined to create new perfect codes.

New lower bounds on the number of q-ary perfect codes are established.

Abstract

The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $\mu$-components $K_\mu$, where $\mu$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\mu$-components, and new lower bounds on the number of perfect 1-error-correcting $q$-ary codes are presented.