Closed Expressions for the Weight Distributions of Codes Associated with Perfect Codes

TL;DR

This paper derives explicit weight distributions for five related families of codes, including perfect and nearly perfect codes, using methods that avoid complex tools.

Contribution

It provides the first complete weight distributions for these five code families with simplified methods, unifying their analysis.

Findings

Complete weight distributions for all five code families.

Methods avoid heavy computational tools.

Unification of related code structures analysis.

Abstract

Perfect codes are arguably the most fascinating structures in combinatorial coding theory, and their classification and weight distribution are of considerable interest. This classification also involves the analysis of some related structures. This paper considers five closely related structures, but all of them have never been tied together before. These structures are 1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and one family of completely regular codes (to be called diamond codes). The current work concentrates on the weight distributions of these five families of codes. In the past, some of these weight distributions were not computed, some required heavy tools, and for some only the weight enumerator was presented. We provide complete weight distributions for all five families using some methods that do not require any heavy tools.