On extended 1-perfect bitrades

TL;DR

This paper explores extended 1-perfect bitrades in Hamming schemes, establishing their multiple equivalent definitions, existence conditions for certain parameters, and nonexistence results for odd lengths.

Contribution

It introduces five equivalent definitions of extended 1-perfect bitrades and determines their existence and nonexistence conditions based on code parameters.

Findings

Existence of extended 1-perfect bitrades if and only if n=lq+2 for q=2^m.

Nonexistence of such bitrades when n is odd.

Multiple equivalent characterizations of extended 1-perfect bitrades.

Abstract

Extended $1$-perfect codes in the Hamming scheme $H(n,q)$ can be equivalently defined as codes that turn to $1$-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-$4$ codes with certain dual distances. We define extended $1$-perfect bitrades in $H(n,q)$ in five different manners, corresponding to the different definitions of extended $1$-perfect codes, and prove the equivalence of these definitions of extended $1$-perfect bitrades. For $q=2^m$, we prove that such bitrades exist if and only if $n=lq+2$. For any $q$, we prove the nonexistence of extended $1$-perfect bitrades if $n$ is odd. Keywords: Perfect code, Extended perfect code, Bitrade, Completely regular code, Uniformly packed code.