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Register a new userOn Cube Tilings of Tori and Classification of Perfect Codes in the Maximum Metric
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We describe odd-length-cube tilings of the n-dimensional q-ary torus what includes q-periodic integer lattice tilings of R^n. In the language of coding theory these tilings correspond to perfect codes with respect to the maximum metric. A complete characterization of the two-dimensional tillings is presented and in the linear case, a description of general matrices, isometry and isomorphism classes is provided. Several methods to construct perfect codes from codes of smaller dimension or via sections are derived. We introduce a special type of matrices (perfect matrices) which are in correspondence with generator matrices for linear perfect codes in arbitrary dimensions. For maximal perfect codes, a parametrization obtained allows to describe isomorphism classes of such codes. We also approach the problem of what isomorphism classes of abelian groups can be represented by q-ary n-dimensional perfect codes of a given cardinality N.
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We investigate perfect codes in $mathbb{Z}^n$ under the $ell_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n = 2, 3$, we determine all radii for which there are linear perfect codes. The non-existence results for codes in $mathbb{Z}^n$ presented here imply non-existence results for codes over finite alphabets $mathbb{Z}_q$, when the alphabet size is large enough, and has implications on some recent constructions of spherical codes.
Perfect truncated-metric codes (PTMCs) in the $n$-dimensio-nal grid $Lambda_n$ of $mathbb{Z}^n$ ($0<ninmathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance $rho(u,v)$ in $mathbb{Z}^n$ given between vertices $u=(u_1,cdots,u_n)inmathbb{Z}^n$ and $v=(v_1, ldots,v_n)inmathbb{Z}^n$ as the Hamming distance $h(u,v)$ in $mathbb{Z}^n$ (or graph distance $h(u,v)$ in $Lambda_n$) if $|u_i-v_i|le 1$, for all $iin{1, ldots,n}$, and as $n+1$, otherwise. While this $rho$ is related to the $ell_p$ metrics, PTMCs associated with lattice tilings of $mathbb{Z}^n$ were recently worked upon as rainbow perfect dominating sets. Now, PTMCs are extended to ternary compounds $Gamma_n$ obtained by glueing, or locking, ternary $n$-cubes along their codimension 1 ternary subcubes. Such compounds may be taken as alternate reality graphs, since they offer a third option in each coordinate direction at each vertex. We ascertain the existence of an infinite number of isolated PTMC of radius 2 in $Gamma_n$ for $n=2$ and conjecture such existence for $n>2$ with radius $n$. We ask whether there exists a suitable notion replacing that of quotient toroidal grids of $Lambda_n$ for the case of $Gamma_n$.
Let $d, n in mathbb{Z}^+$ such that $1leq d leq n$. A $d$-code $mathcal{C} subset mathbb{F}_q^{n times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively of symmetric, alternating and hermitian matrices. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric $2$-code which is not equivalent to the one with same parameters known so far.
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)leq F(G)leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for bipartite graphs $G$, $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. In this paper, we solve the problem for general graphs and obtain that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or $K^+_{n,n}$ ($K_{n,n}$ with arbitrary additional edges in the same partite set). For a larger class of graphs $G$ with $F(G)=n-1$ we show that $G$ is $n$-connected and a brick (3-connected and bicritical graph) except for $K^+_{n,n}$. In particular, we prove that the forcing spectrum of each such graph $G$ is continued by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $lfloorfrac{n}{2}rfloor$ to $n-1$.
We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.
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