On rank and kernel of some mixed perfect codes
Let \(Z(n,l)\) denote the direct product \(Z_2\times\cdots\times Z_2\times Z_2^l\), where there are exactly \(n\) factors \(Z_2\) preceding \(Z_2^l\). A word \(\bar{t}\in Z(n,l)\) is a tuple of type \((t_1,\dots,t_n,(t_{n+1},\dots,t_{n+l}))\), where the \(l\)-tuple \((t_{n+1},\dots,t_{n+l})\) is considered as one object instead of consisting of \(l\) bits. The \textit{distance} \(d\) between two words \(\bar{c},\bar{c}'\in Z(n,l)\) is the number of positions in which \(\bar{c}\) and \(\bar{c}'\) differ in the first \(n\) positions, plus one if \(\bar{c}\) and \(\bar{c}'\) are not equal in the \(l\)-tuple at the end of \(\bar{c}\) and \(\bar{c}'\). A \textit{perfect 1-error correcting code} or \textit{mixed perfect code }in \(Z(n,l)\) is a subset \(C\) of \(Z(n,l)\) such that for every \(\bar{x}\in Z(n,l)\), there is a unique \(\bar{c}\in Z(n,l)\) satisfying \(d(\bar{x},\bar{c})\leq 1\). For \(l=0\), this reduces to the classical definition of a binary perfect 1-error correcting code. The \textit{kernel} ker\((C)\) of a subset \(C\subseteq Z(n,l)\) is the set of vectors \(\bar{p}\in Z(n,l)\) for which \(C+\bar{p}=C\). There is a vast literature on binary perfect 1-error correcting codes in \(Z_2^n\). In this article, the authors discuss mixed perfect codes in \(Z(n,l)\). As concrete results, they give a lower bound and an upper bound for the rank \(k\) of the kernel of a mixed perfect code in \(Z(n,l)\) depending on the rank \(r\) of the mixed perfect code, and they give two general constructions for mixed perfect codes in \(Z(n,l)\). The first construction gives a linear mixed perfect code and the second construction gives a non-linear mixed perfect code. As an interesting feature of the linear mixed perfect codes obtained via Construction 1, the sharpness of one of the upper bounds for the rank \(k\) of the kernel of a mixed perfect code is proven.
scientific article; zbMATH DE number 5644066