New nonexistence results on perfect permutation codes under the hamming metric

<p style='text-indent:20px;'>Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in <inline-formula><tex-math id="M1">\begin{document}$ S_n $\end{document}</tex-math></inline-formula>, the set of all permutations on <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> elements, under the Hamming metric. We prove the nonexistence of perfect <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting codes in <inline-formula><tex-math id="M4">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric, for more values of <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect <inline-formula><tex-math id="M7">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting code in <inline-formula><tex-math id="M8">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric for some <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ t = 1,2,3,4 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M11">\begin{document}$ 2t+1\leq n\leq \max\{4t^2e^{-2+1/t}-2,2t+1\} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M12">\begin{document}$ t\geq 2 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M13">\begin{document}$ \min\{\frac{e}{2}\sqrt{n+2},\lfloor\frac{n-1}{2}\rfloor\}\leq t\leq \lfloor\frac{n-1}{2}\rfloor $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ n\geq 7 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M15">\begin{document}$ e $\end{document}</tex-math></inline-formula> is the Napier's constant.</p>

Codes from incidence matrices of some regular graphs

We examine the [Formula: see text]-ary linear codes with respect to Lee metric from incidence matrix of the Lee graph with vertex set [Formula: see text] and two vertices being adjacent if their Lee distance is one. All the main parameters of the codes are obtained as [Formula: see text] if [Formula: see text] is odd and [Formula: see text] if [Formula: see text] is even. We examine also the [Formula: see text]-ary linear codes with respect to Hamming metric from incidence matrices of Desargues graph, Pappus graph, Folkman graph and the main parameters of the codes are [Formula: see text], respectively. Any transitive subgroup of automorphism groups of these graphs can be used for full permutation decoding using the corresponding codes. All the above codes can be used for full error correction by permutation decoding.

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

The Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

Formulas for a characteristic of spheres and balls in binary high-dimensional spaces

We consider a special function ρ(H) of the subset H of n-dimensional vector linear space over the field K. This function is used in the estimates of accuracy of the Poisson approximation for the distribution of the number of solutions of systems of random equations and random inclusions over K. For the case when K = GF(2) and H is a sphere or ball (in the Hamming metric) in {0, 1}n we obtain explicit and approximate formulas for ρ(H) for sufficiently large values of n.

Perfect codes in poset block spaces

There is a limited class of perfect codes with respect to the classical Hamming metric. There are other kind of metrics with respect to which perfect codes have been investigated viz. poset metric, block metric and poset block metric. Given the minimal elements of a poset, a necessary and sufficient condition for [Formula: see text]-perfectness of a poset block code has been derived. A necessary and sufficient condition for a poset block code to be [Formula: see text]-perfect has also been considered. Further, for each [Formula: see text], [Formula: see text], a sufficient condition that ensures the existence of a poset block structure which turns a given code into an [Formula: see text]-perfect poset block code has been obtained. Several illustrations of well known codes to be [Formula: see text]-perfect for specific values of [Formula: see text] have been explored.