On extended 1-perfect bitrades
Evgeny A. Bespalov, Denis S. Krotov
TL;DR
This work defines extended $1$-perfect bitrades in the Hamming graph $H(n,q)$ through five equivalent formulations that mirror the corresponding five definitions for extended $1$-perfect codes. It establishes the equivalence of these bitrade definitions, derives necessary conditions (e.g., $n$ even and $n=lq+2$ when $q=2^m$), and proves nonexistence for odd $n$ across all $q$. The existence results for $q=2^m$ are constructed via extendable spherical bitrades and tensor-product techniques, tying the theory of bitrades to diameter-perfect, completely regular, and uniformly packed code frameworks. Overall, the paper extends the trades paradigm to extended code families in $H(n,q)$, providing a unified view and concrete parametric constraints for existence and nonexistence.
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.