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Register a new userAn Assmus-Mattson Theorem for Rank Metric Codes
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Added by
Eimear Byrne
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criteria for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring subcodes of Gabidulin codes, if each of the rightmost $delta-1$ columns in Ferrers diagram $cal F$ has at least $n-r$ dots, where $r$ is taken in a range, then the conditions that an FDRM code in $cal F$ is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.
Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field $mathbb F_{q^m}$ (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and parameters.
In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. We apply the various bounds to the classes of maximal rank distance and quasi maximal rank distance codes.
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