154,178 research outputs found

    Partial ovoids and partial spreads in finite classical polar spaces

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    We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces

    Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

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    We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]

    Maximal partial line spreads of non-singular quadrics

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    For n >= 9 , we construct maximal partial line spreads for non-singular quadrics of for every size between approximately and , for some small constants and . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gacs and SzAnyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles and by Pepe, Roing and Storme

    Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

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    In this paper we investigate partial spreads of H(2n1,q2)H(2n-1,q^2) through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of H(2n1,q2)H(2n-1,q^2). We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of H(3,q2)H(3,q^2) for a range of sizes

    Partial Spreads in Random Network Coding

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    Following the approach by R. K\"otter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F_q^n, q being a prime power and F_q the finite field with q elements. In particular, following an idea in finite projective geometry, we introduce a class of network codes which we call "partial spread codes". Partial spread codes naturally generalize spread codes. In this paper we provide an easy description of such codes in terms of matrices, discuss their maximality, and provide an efficient decoding algorithm

    A geometric proof of the upper bound on the size of partial spreads in H(4n+1, q²)

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    We give a geometric proof of the upper bound of q(2n+1) + 1 on the size of partial spreads in the polar space H(4n + 1, q(2)). This bound is tight and has already been proved in an algebraic way. Our alternative proof also yields a characterization of the partial spreads of maximum size in H(4n + 1, q(2))

    A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd

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    In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a approximate to q2 and b approximate to 9/10q2, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle IN(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadrangle Q(4, q), the same result is obtained for maximal partial spreads of 1N(q), q odd. This article concludes a series of articles on spectrum results on maximal partial ovoids of Q(4, q), on spectrum results on maximal partial spreads of VV(q), on spectrum results on maximal partial 1-systems of Q(+)(5,q), and on spectrum results on minimal blocking sets with respect to the planes of PG(3, q). We conclude this article with the tables summarizing the results
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