On extended perfect codes

TL;DR

This work fully characterizes the existence of extended $1$-perfect codes in Hamming graphs $H(n,q)$ when $q$ is a prime power, showing such codes occur only in the three families $H(2,q)$, $H(q+2,q)$ with $q=2^m$, and $H(2^k,2)$. The authors derive this classification via an analysis of equitable distance partitions, their quotient matrices, and connections to weight distributions and Krawtchouk polynomials, introducing $S'=rac{n(q-1)E-S}{q}$ and exploiting its Jordan form. Furthermore, the paper obtains new necessary conditions for the existence of $2$-perfect codes in $H(n,q)$ and in $J(n,w)$ by examining the corresponding quotient matrices and eigenstructures, which constrain possible parameter sets even when extended $1$-perfect codes exist. These results sharpen the understanding of perfect-code existence in these graph families and provide tools to rule out large families of parameters, while leaving the non-prime-power case open for future work.

Abstract

We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for $q=3$, $4$, $n>q+2$. In this work, we characterize all positive integers $n$, $r$ and prime $p$, for which there exist such a code in $H(n,p^r)$. We also consider $2$-perfect codes in Hamming $H(n,q)$ and Johnson graphs $J(n,w)$ and find new necessary conditions on there existence.

Theorems & Definitions (31)

• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 2
• Theorem 1
• proof