5,823 research outputs found

    Codes and caps from orthogonal Grassmannians

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    In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding εkgr\varepsilon_k^{gr} of an orthogonal Grassmannian Δk\Delta_k. In particular, we determine some of the parameters of the codes arising from the projective system determined by εkgr(Δk)\varepsilon_k^{gr}(\Delta_k). We also study special sets of points of Δk\Delta_k which are met by any line of Δk\Delta_k in at most 2 points and we show that their image under the Grassmann embedding εkgr\varepsilon_k^{gr} is a projective cap.Comment: Keywords: Polar Grassmannian; dual polar space; embedding; error correcting code; cap; Hadamard matrix; Sylvester construction (this is a slightly revised version of v2, with updated bibliography

    A characterization of MDS codes that have an error correcting pair

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    Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were found independently by R. K\"otter (1992), as a general algebraic method of decoding linear codes. These pairs exist for several classes of codes. However little or no study has been made for characterizing those codes. This article is an attempt to fill the vacuum left by the literature concerning this subject. Since every linear code is contained in an MDS code of the same minimum distance over some finite field extension we have focused our study on the class of MDS codes. Our main result states that an MDS code of minimum distance 2t+12t+1 has a tt-ECP if and only if it is a generalized Reed-Solomon code. A second proof is given using recent results Mirandola and Z\'emor (2015) on the Schur product of codes

    Coding Theory and Algebraic Combinatorics

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    This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201
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