1,721,204 research outputs found
On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric
No full textWe discuss the functional codes C-h(Q(N)), for small h >= 3, q > 9, and for N >= 6. This continues the study of different classes of functional codes, performed on functional codes arising from quadrics and Hermitian varieties. Here, we consider the functional codes arising from the intersections of the algebraic hypersurfaces of small degree h with a given non-singular quadric Q(N) in PG(N, q)
The second and the third smallest arrangements of hyperplanes in finite projective spaces
Full text linkIn this paper we determine for some values of d the second and the third smallest configuration of hyperplanes in PG(N, q). We present links with the unique extendability of arcs in PG(2, q) and with (k, 3)-arcs having a unique trisecant
Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes
No full textWe present bounds on the number of points in algebraic curves and algebraic hypersurfaces in P-n(F-q) of small degree d, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have applications to the weight distribution of the projective Reed-Muller codes PRM(q, d, n) over the finite field IFq
Maximal sets of k-spaces pairwise intersecting in at least a (k-2)-space
No full textIn this paper, we analyze the structure of maximal sets of k-dimensional spaces in PG(n, q) pairwise intersecting in at least a (k - 2)-dimensional space, for 3 <= k <= n - 2. We give an overview of the largest examples of these sets with size more than f(k, q) = max{3q(4) + 6q(3) + 5q(2) + q + 1,theta(k+1) + q(4) + 2q(3) + 3q(2)}
On codewords in the dual code of classical generalised quadrangles and classical polar spaces
No full textAbstractIn [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73–92], the codewords of small weight in the dual code of the code of points and lines of Q(4,q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q+(5,q) and H(5,q2), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q+(5,q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4,q), q even. To prove this, we show that a blocking set of Q(4,q), q even, of size q2+1+r, where 0<r<(q+4)/6, contains an ovoid of Q(4,q), improving on [J. Eisfeld, L. Storme, T. Szőnyi, P. Sziklai, Covers and blocking sets of classical generalised quadrangles, Discrete Math. 238 (2001) 35–51, Theorem 9]
A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q odd
Full text linkIn 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
On sets of subspaces with two intersection dimensions and a geometrical junta bound
No full textIn this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a junta (Combin Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families
Theorems of Erdos-Ko-Rado type in polar spaces
Full text linkAbstractWe consider Erdős–Ko–Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erdős–Ko–Rado sets of generators of maximum size in all polar spaces, except for H(4n+1,q2) with n⩾2
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