529 research outputs found

    Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

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    The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table

    Generalized vector space partitions

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    A vector space partition P\mathcal{P} in Fqv\mathbb{F}_q^v is a set of subspaces such that every 11-dimensional subspace of Fqv\mathbb{F}_q^v is contained in exactly one element of P\mathcal{P}. Replacing "every point" by "every tt-dimensional subspace", we generalize this notion to vector space tt-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case q=1q=1.Comment: 12 pages, typos correcte

    Subspace Packings : Constructions and Bounds

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    The Grassmannian Gq(n,k)\mathcal{G}_q(n,k) is the set of all kk-dimensional subspaces of the vector space Fqn\mathbb{F}_q^n. K\"{o}tter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are qq-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian Gq(n,k)\mathcal{G}_q(n,k) also form a family of qq-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of qq-analogs of block designs which was not considered before. This family, called subspace packings, is the qq-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing tt-(n,k,λ)q(n,k,\lambda)_q is a set S\mathcal{S} of kk-subspaces from Gq(n,k)\mathcal{G}_q(n,k) such that each tt-subspace of Gq(n,t)\mathcal{G}_q(n,t) is contained in at most λ\lambda elements of S\mathcal{S}. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.Comment: 30 pages, 27 tables, continuation of arXiv:1811.04611, typos correcte

    Load-Balanced Fractional Repetition Codes

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    We introduce load-balanced fractional repetition (LBFR) codes, which are a strengthening of fractional repetition (FR) codes. LBFR codes have the additional property that multiple node failures can be sequentially repaired by downloading no more than one block from any other node. This allows for better use of the network, and can additionally reduce the number of disk reads necessary to repair multiple nodes. We characterize LBFR codes in terms of their adjacency graphs, and use this characterization to present explicit constructions LBFR codes with storage capacity comparable existing FR codes. Surprisingly, in some parameter regimes, our constructions of LBFR codes match the parameters of the best constructions of FR codes

    A family of optimal locally recoverable codes

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    A code over a finite alphabet is called locally recoverable (LRC) if every symbol in the encoding is a function of a small number (at most rr) other symbols. We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality. The codewords are obtained as evaluations of specially constructed polynomials over a finite field, and reduce to a Reed-Solomon code if the locality parameter rr is set to be equal to the code dimension. The size of the code alphabet for most parameters is only slightly greater than the code length. The recovery procedure is performed by polynomial interpolation over rr points. We also construct codes with several disjoint recovering sets for every symbol. This construction enables the system to conduct several independent and simultaneous recovery processes of a specific symbol by accessing different parts of the codeword. This property enables high availability of frequently accessed data ("hot data").Comment: Minor changes. This is the final published version of the pape

    Tables of subspace codes

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    One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least dd over Fqn\mathbb{F}_q^n, where the dimensions of the codewords, which are vector spaces, are contained in K⊆{0,1,…,n}K\subseteq\{0,1,\dots,n\}. In the special case of K={k}K=\{k\} one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot
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