6 research outputs found

    Lifted codes and the multilevel construction for constant dimension codes

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    Constant dimension codes are e.g. used for error correction and detection in random linear network coding, so that constructions for these codes have achieved wide attention. Here, we improve over 150 lower bounds by describing better constructions for subspace distance 4.Comment: 40 pages, 8 table

    The interplay of different metrics for the construction of constant dimension codes

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    A basic problem for constant dimension codes is to determine the maximum possible size Aq(n,d;k)A_q(n,d;k) of a set of kk-dimensional subspaces in Fqn\mathbb{F}_q^n, called codewords, such that the subspace distance satisfies dS(U,W):=2k2dim(UW)dd_S(U,W):=2k-2\dim(U\cap W)\ge d for all pairs of different codewords UU, WW. Constant dimension codes have applications in e.g.\ random linear network coding, cryptography, and distributed storage. Bounds for Aq(n,d;k)A_q(n,d;k) are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases Aq(10,4;5)A_q(10,4;5), Aq(11,4;4)A_q(11,4;4), Aq(12,6;6)A_q(12,6;6), and Aq(15,4;4)A_q(15,4;4). We also derive general upper bounds for subcodes arising in those constructions.Comment: 19 pages; typos correcte

    Bounds for the multilevel construction

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    One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space Pq(n)\mathcal{P}_q(n) for a given minimum distance. The determination of the exact maximum cardinality is a very tough discrete optimization problem involving a huge number of symmetries. Besides some explicit constructions for \textit{good} subspace codes several of the most success full constructions involve the solution of discrete optimization subproblems itself, which mostly have not been not been solved systematically. Here we consider the multilevel a.k.a.\ Echelon--Ferrers construction and given lower and upper bounds for the achievable cardinalities. From a more general point of view, we solve maximum clique problems in weighted graphs, where the weights can be polynomials in the field size qq.Comment: 95 page

    Advanced and current topics in coding theory

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