On Nearly Perfect Covering Codes

TL;DR

This paper studies nearly perfect covering codes with covering radius one, classifies them into families, characterizes their properties, and explores their constructions and extended versions, advancing understanding of optimal covering code structures.

Contribution

It introduces a classification of nearly perfect covering codes with radius one, characterizes their properties, and provides constructions and analysis of their extended forms.

Findings

Codes can be partitioned into three families based on minimum distance.

Some code families are fully characterized.

Extended codes reveal unexpected equivalence classes.

Abstract

Nearly perfect packing codes are those codes that meet the Johnson upper bound on the size of error-correcting codes. This bound is an improvement to the sphere-packing bound. A related bound for covering codes is known as the van Wee bound. Codes that meet this bound will be called nearly perfect covering codes. In this paper, such codes with covering radius one will be considered. It will be proved that these codes can be partitioned into three families depending on the smallest distance between neighboring codewords. Some of the codes contained in these families will be completely characterized. Other properties of these codes will be considered too. Construction for codes for each such family will be presented, the weight distribution and the distance distribution of codes from these families are characterized. Finally, extended nearly perfect covering code will be considered and unexpected equivalence classes of codes of the three types will be defined based on the extended codes.