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Register a new userPerfect codes in the lp metric
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Added by
Antonio Campello
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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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.
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A $t$-$(n,d,lambda)$ design over ${mathbb F}_q$, or a subspace design, is a collection of $d$-dimensional subspaces of ${mathbb F}_q^n$, called blocks, with the property that every $t$-dimensional subspace of ${mathbb F}_q^n$ is contained in the same number $lambda$ of blocks. A collection of matrices in over ${mathbb F}_q$ is said to hold a subspace design if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a subspace design.
A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger length
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.
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 $Gamma_n$.
How can $d+k$ vectors in $mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $theta(d,k):=min_Xmax_{x eq yin X}|langle x,yrangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $Xsubseteqmathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $dtoinfty$. In establishing bounds on $theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1choose 2}$ equiangular lines in $mathbb{R}^k$. Using this connection, we are able to pin down $theta(d,k)$ whenever $kin{1,2,3,7,23}$ and establish asymptotics for general $k$. The main tool is an upper bound on $mathbb{E}_{x,ysimmu}|langle x,yrangle|$ whenever $mu$ is an isotropic probability mass on $mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $mathbb{C}^k$, also known as SIC-POVM in physics literature.
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