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1.
2.
SL(n, q) is the group of n×n matrices, over the Galois field GF(q), of determinate 1. PSL(n, q) is SL(n, q) modulo the scalar n×n matrices of determinate 1. PSL(n, q) acts on the Desarguesian projective space PG(n−1, q). Sp(4, q) is the group of 4 × 4 matrices of determinate 1 which preserve the symplectic bilinear form on the 4 × 1 matrices over GF(q). PSp(4, q) is Sp(4, q) modulo Z = {1,−1}. PSp(4, q) acts on the symplectic generalized quadrangle W(3, q), a subspace of the projective space PG(3, q), as a group of automorphisms. In this paper, bounds are given for the genus of these groups. 相似文献
3.
This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q
2-1-r lines, then r=s(
+1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,
). We also discuss maximal partial spreads in PG(3,p
3), p=p
0
h
, p
0 prime, p
0 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p
2+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p
3) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p
3). In PG(3,p
3),p square, for maximal partial spreads of deficiency p
2+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1,
), which implies that 0 (mod
+1) and, when (2t+1)(
-1) <q-1, that 2(
+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,
) and this implies 0 (mod q
2/3+q
1/3+1). A more general result is also presented. 相似文献
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5.
Vito Napolitano 《Journal of Geometry》2014,105(3):449-455
A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q 2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given. 相似文献
6.
Letm′2(3,q) be the largest value ofk(k<q 2+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′2(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4. 相似文献
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István Kovács 《Designs, Codes and Cryptography》2001,24(1):37-42
In PG(2,q) it is well known that if k is close to q, then any k-arc is contained in a conic. The internal nuclei of a point set form an arc. In this article it is proved that for q odd the above bound on the number of points could be lowered to
(or even less), if the arc is obtained as the set of internal nuclei of some point set of proper size. Using this result the internal nuclei of point sets of size q + 1 will be studied in higher dimensional spaces, and an application will be presented to so-called threshold schemes. 相似文献
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10.
Let Ω and
be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and
. Denote by K the cone of vertex Ω and base
and consider the point set B defined by
in the André, Bruck-Bose representation of PG(2,qn) in PG(2n,q) associated to a regular spread
of PG(2n−1,q). We are interested in finding conditions on
and Ω in order to force the set B to be a minimal blocking set in PG(2,qn) . Our interest is motivated by the following observation. Assume a Property α of the pair (Ω,
) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of
all known pairs (Ω,
) with Property α. With this in mind, we deal with the problem in the case Ω is a subspace of PG(2n−1,q) and
a blocking set in a subspace of PG(2n,q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2,qn), generalizing the main constructions of [14]. For example, for q = 3h, we get large blocking sets of size qn + 2 + 1 (n≥ 5) and of size greater than qn+2 + qn−6 (n≥ 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2,q2k) is also given. 相似文献
11.
Helmut Völklein 《Israel Journal of Mathematics》1993,82(1-3):405-427
We determine the braid group action on generating systems of a group that is the semi-direct product of a finite vector space
with a group of scalars. This leads to Galois realizations of certain groups GL
n
(q) and PU
n
(q).
Dedicated to Prof. J. G. Thompson
Partially supported by an NSA grant.
This work was done while the author was a fellow of the Institute for Advanced Studies in Jerusalem. 相似文献
12.
In this paper we construct maximal partial spreads in PG(3, q) which are a log q factor larger than the best known lower bound. For n ≥ 5 we also construct maximal partial spreads in PG(n, q) of each size between cnq n ? 2 log q and c′ q n ? 1. 相似文献
13.
J. D. Key 《Geometriae Dedicata》1975,4(2-4):377-386
14.
Christiane Lefevre-Percsy 《Geometriae Dedicata》1982,12(2):209-213
This paper is a first step in the characterization of finite Hermitian quadrics as sets of class (0, 1, n, q+1) in PG(d, q). 相似文献
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16.
A (k,n)-arc in PG(2,q) is usually defined to be a set
of k points in the plane such that some line meets
in n points but such that no line meets
in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow
to be a multiset, that is, permit
to contain multiple points. The case k=q
2+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q
2+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q
2+q+2 and minimum distance q
2. Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q
2+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem. 相似文献
17.
The study of the intersection of two Baer subgeometries of PG(n, q), q a square, started in Bose et al. (Utilitas Math 17, 65–77, 1980); Bruen (Arch Math 39(3), 285–288, (1982). Later, in Svéd (Baer subspaces in the n-dimensional projective space. Springer-Verlag) and Jagos et
al. (Acta Sci Math 69(1–2), 419–429, 2003), the structure of the intersection of two Baer subgeometries of PG(n, q) has been completely determined. In this paper, generalizing the previous results, we determine all possible intersection
configurations of any two subgeometries of PG(n, q).
相似文献
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Functional Analysis and Its Applications - 相似文献
20.
Alexander A. Davydov Giorgio Faina Stefano Marcugini Fernanda Pambianco 《Journal of Geometry》2009,94(1-2):31-58
More than thirty new upper bounds on the smallest size t 2(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t 2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m 2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m 2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t 2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701. 相似文献