共查询到20条相似文献,搜索用时 31 毫秒
1.
G. L. Ebert (1985) constructed (qn + 1)-caps in PG(2n − 1, q), n even, which were the orbits of the subgroup of order qn + 1 of a cyclic Singer group of PG(2n − 1, q). This article shows that these caps are the intersection of n − 1 linearly independent elliptic quadrics of PG(2n − 1, q). 相似文献
2.
Generalized Hadamard matrices of order qn−1 (q—a prime power, n2) over GF(q) are related to symmetric nets in affine 2-(qn,qn−1,(qn−1−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q). 相似文献
3.
We define a mixed partition of Π = PG(d, q r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q 2) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q 2) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q 2n . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r 2 n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q rn ). 相似文献
4.
In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r2(q) be the number such that q+r2(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most qm+qm−1+…+q+1+r2(q)·(∑i=2m−n′m−1qi) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals qm+qm−1+…+q+1+r2(q)(∑i=2m−n′m−1qi). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<qm+qm−1+…+q+1+r2(q)(qm−1+qm−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if
, then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. For q=p3h, p7 and q not a square we show this assertion for |B|qm+qm−1+…+q+1+q2/3·(qm−1+…+1). 相似文献
5.
David Paget 《Journal of Approximation Theory》1988,54(3)
Let f ε Cn+1[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials qk associated with a distribution dα on [−1, 1]. It is shown that if qn+1/qn max(qn+1(1)/qn(1), −qn+1(−1)/qn(−1)), then f − H[f] fn + 1 · qn+1/qn + 1(n + 1), where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)α (1 + t)β dt, α, β > −1, the condition on qn+1/qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2. 相似文献
6.
Ronald D. Baker Jeremy M. Dover Gary L. Ebert Kenneth L. Wantz 《Journal of Geometry》2000,67(1-2):23-34
For any odd integern 3 and prime powerq, it is known thatPG(n–1, q2) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n–1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n–1, q2). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes. 相似文献
7.
Let n and k be positive integers. Let Cq be a cyclic group of order q. A cyclic difference packing (covering) array, or a CDPA(k, n; q) (CDCA(k, n; q)), is a k × n array (aij) with entries aij (0 ≤ i ≤ k−1, 0 ≤ j ≤ n−1) from Cq such that, for any two rows t and h (0 ≤ t < h ≤ k−1), every element of Cq occurs in the difference list
at most (at least) once. When q is even, then n ≤ q−1 if a CDPA(k, n; q) with k ≥ 3 exists, and n ≥ q+1 if a CDCA(k, n; q) with k ≥ 3 exists. It is proved that a CDCA(4, q+1; q) exists for any even positive integers, and so does a CDPA(4, q−1; q) or a CDPA(4, q−2; q). The result is established, for the most part, by means of a result on cyclic difference matrices with one hole, which is of interest in its own right. 相似文献
8.
The structure of
n– 1-sets inPG(n, q) with more thanq – 1 nuclei is investigated. It is shown that classification of these sets with the maximal numberq
n– 1-q
n– 2 of nuclei is equivalent to the classification of (q + l)-sets inPG(2,q) havingq –1 nuclei.Dedicated to Professer Walter Benz for his 60th birthday 相似文献
9.
In this paper we introduce semi-pseudo-ovoids, as generalizations of the semi-ovals and semi-ovoids. Examples of these objects are particular classes of SPG-reguli and some classes of m-systems of polar spaces. As an application it is proved that the axioms of pseudo-ovoid O(n,2n,q) in PG(4n−1,q) can be considerably weakened and further a useful and elegant characterization of SPG-reguli with the polar property is given.Research Assistant of the Fund for Scientific Research Flanders (FWO-Vlaanderen) 相似文献
10.
Guglielmo Lunardon 《Geometriae Dedicata》1999,75(3):245-261
In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t – 1, q) into the Grassmannian of the subspaces of rank t of PG(2t – 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt – 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt – 1, q) by a cap V
r, t
of PG(r
t
– 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r – 1), and that the collineation group of PG(r
t
– 1, q) stabilizing V
r, t
acts 2-transitively on V
r, t
. In particular, we prove that V
3, 2 is the union of q
2 – q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q
2) is represented by a hyperplane section U of V
3, 2. For q 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q
+ (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q
2). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino. 相似文献
11.
12.
We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qn−1/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28. 相似文献
13.
The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s
2 by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s
and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q
3) over GF(q), and on a representation of GF(q)3 in GF(q
3)3 indicated in Jamison [4]. 相似文献
14.
A new extension theorem for linear codes 总被引:1,自引:0,他引:1
For an [n,k,d]q code
with k3, gcd(d,q)=1, the diversity of
is defined as the pair (Φ0,Φ1) withAll the diversities for [n,k,d]q codes with k3, d−2 (mod q) such that Ai=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated. 相似文献
15.
Joseph E. Bonin 《Advances in Applied Mathematics》1996,17(4):460-476
It is known that a geometry with rankrand no minor isomorphic to the (q+2)-point line has at most (qr−1)/(q−1) points, with strictly fewer points ifr>3 andqis not a prime power. Forqnot a prime power andr>3, we show thatqr−1−1 is an upper bound. Forqa prime power andr>3, we show that any rank-rgeometry with at leastqr−1points and no (q+2)-point-line minor is representable overGF(q). We strengthen these bounds toqr−1−(qr−2−1)/(q−1)−1 andqr−1−(qr−2−1)/(q−1) respectively whenqis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor. 相似文献
16.
We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (qr − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice. 相似文献
17.
Tatsuya Maruta 《Geometriae Dedicata》1995,54(3):263-266
Existence and uniqueness of pseudo-cyclic [q
2+1,q
2–3, 4]-codes over GF(q) are proved. Elliptic quadrics are characterized as those (q
2+1)-caps in PG(3,q) whose corresponding [q
2+1,q
2–3, 4]-codes are pseudo-cyclic. 相似文献
18.
Primitive polynomial with three coefficients prescribed 总被引:1,自引:1,他引:0
The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a1,a2,a3)Fq3, there exists a primitive polynomial f(x)=xn−σ1xn−1++(−1)nσn over Fq of degree n with the first three coefficients σ1,σ2,σ3 prescribed as a1,a2,a3 when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over Fq with the first three coefficient prescribed where the characteristic of Fq is 2 or 3. 相似文献
19.
We classify all embeddings θ: PG(n, q) → PG(d, q), with $d \geqslant \tfrac{{n(n + 3)}}
{2}$d \geqslant \tfrac{{n(n + 3)}}
{2}, such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ?n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well. 相似文献
20.
J. R. M. Mason 《Geometriae Dedicata》1984,15(4):355-361
A (k, d)-arc in PG(2, q) is a set of k points such that some d, but no d+1, of them are collinear. An outstanding problem is to find the maximum value of k for which a (k, d)-arc exists. A construction is given for a class of (k, p
n–p
m)-arcs in PG(2, p
n). These arcs constitute a lower bound on the maximum possible value of k, and a subset of them is shown to be optimal. 相似文献