Perfect codes from the dual point of view. I (Q998352)

In this paper \(1\)-error correcting binary perfect codes are considered. The author generalizes the concepts of dual code and parity check matrix for non-linear cases and shows how to use this generalization for the enumeration and classification of some particular classes of perfect \(1\)-error correcting binary codes. A sample of statements proved: \begin{itemize} \item[(i)] There is exactly one equivalence class of perfect codes of length \(31\), rank \(30\) and with dimension of kernel \(23\). \item[(ii)] There are exactly three equivalence classes of perfect codes of length \(15\), rank \(12\) and with dimension of kernel \(9\). \item[(iii)] There exists a perfect code \(C\) such that there is no Hamming code \(H\) with \[ \ker(C)\subseteq H \subseteq(C)^{\perp}. \] \end{itemize}