43,271 research outputs found

    Equidistant Codes in the Grassmannian

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    Equidistant codes over vector spaces are considered. For kk-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker embedding, for 1-intersecting codes of kk-dimensional subspaces over \F_q^n, nβ‰₯(k+12)n \geq \binom{k+1}{2}, where the code size is qk+1βˆ’1qβˆ’1\frac{q^{k+1}-1}{q-1} is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size nΓ—(n2)n \times \binom{n}{2} over \F_q, rank nβˆ’1n-1, and rank distance nβˆ’1n-1.Comment: 16 page

    Problems on q-Analogs in Coding Theory

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    The interest in qq-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

    Optimal Ferrers Diagram Rank-Metric Codes

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    Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.Comment: to be presented in Algebra, Codes, and Networks, Bordeaux, June 16 - 20, 201
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