On 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.