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1.
Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with
For n = 2 or 3 the characteristic function
of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of
is
for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg(
for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least
相似文献
2.
We transfer the whole geometry of PG(3, q) over a non-singular quadric Q4,q of PG(4, q) mapping suitably PG(3, q) over Q4,q. More precisely the points of PG(3, q) are the lines of Q4,q; the lines of PG(3, q) are the tangent cones of Q4,q and the reguli of the hyperbolic quadrics hyperplane section of Q4,q. A plane of PG(3, q) is the set of lines of Q4,q meeting a fixed line of Q4,q. We remark that this representation is valid also for a projective space
over any field K and we apply the above representation to construct maximal partial spreads
in PG(3, q). For q even we get new cardinalities for
For q odd the cardinalities are partially known. 相似文献
3.
We characterize the finite Veronesean
of all Hermitian varieties of PG(n,q2) as the unique representation of PG(n,q2) in PG(d,q), d n(n+2), where points and lines of PG(n,q2) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q2) generates PG(d,q). Using this result for n=2, we show that
is characterized by the following properties: (1)
; (2) each hyperplane of PG(8,q) meets
in q2+1, q3+1 or q3+q2+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with
shares exactly q2+1 points with it.51E24 相似文献
4.
Martin Hildebrand 《Journal of Algebraic Combinatorics》1992,1(2):133-150
This paper studies a random walk based on random transvections in SL
n(F
q
) and shows that, given
> 0, there is a constant c such that after n + c steps the walk is within a distance
from uniform and that after n – c steps the walk is a distance at least 1 –
from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL
n(F
q
) to compare them with random transvections. 相似文献
5.
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
6.
Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice
, spanning a Euclidean space V. Let m be a positive integer and
be the arrangement of hyperplanes in V of the form
for
and
. It is known that the number
of bounded dominant regions of
is equal to the number of facets of the positive part
of the generalized cluster complex associated to the pair
by S. Fomin and N. Reading.
We define a statistic on the set of bounded dominant regions of
and conjecture that the corresponding refinement of
coincides with the $h$-vector of
. We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ,
orbits of the action of W on the quotient
and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set
of all dominant regions of
. We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1.
2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15 相似文献
7.
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
8.
The peak algebra
is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks.
By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of
. We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak
algebra. We use these bases to describe the Jacobson radical of
and to characterize the elements of
in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals
of
, j = 0,...,
, such that
is the linear span of sums of permutations with a common set of interior peaks and
is the peak algebra. We extend the above results to
, generalizing results of Schocker (the case j = 0).
Aguiar supported in part by NSF grant DMS-0302423
Orellana supported in part by the Wilson Foundation 相似文献
9.
We show that if the number of directions not determined by a pointset
of
, of size q2 is at least pe q then every plane intersects
in 0 modulo pe+1 points and apply the result to ovoids of the generalised quadrangles
and
. 相似文献
10.
The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions
+(q) and
−(q) of (q) to the set
+(q) of secant (hyperbolic) lines and to the set
−(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme
−(q) is pseudocyclic.We further show that the coherent configurations (q
2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme
+(q
2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes
+(q
2) and
−(q
2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new. 相似文献
11.
Hiroaki Taniguchi 《Graphs and Combinatorics》2007,23(4):455-465
Let q = 2l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval
over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qn) with n≥ d + 1 and for any generator σ of the Galois group of GF(qn) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(qd+1), two dual hyperovals
and
in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(qd+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3. 相似文献
12.
Yong Ge TIAN 《数学学报(英文版)》2006,22(1):289-300
For any element a in a generalized 2^n-dimensional Clifford algebra Lln (F) over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln(F) such that Pn^-1DnPn= φ(α)∈F^2n×2n, where φ(a) is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate. 相似文献
13.
Jing YANG Shi Xin LUO Ke Qin FENG 《数学学报(英文版)》2006,22(3):833-844
Assume that m ≥ 2, p is a prime number, (m,p(p - 1)) = 1,-1 not belong to 〈p〉 belong to (Z/mZ)^* and [(Z/mZ)^*:〈p〉]=4.In this paper, we calculate the value of Gauss sum G(X)=∑x∈F^*x(x)ζp^T(x) over Fq,where q=p^f,f=φ(m)/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G(x) belongs to the decomposition field K of p in Q(ζm) and K is an imaginary quartic abelian unmber field.When the Galois group Gal(K/Q) is cyclic,we have studied this cyclic case in anotyer paper:"Gauss sums of index four:(1)cyclic case"(accepted by Acta Mathematica Sinica,2003).In this paper we deal with the non-cyclic case. 相似文献
14.
Ke Qin FENG Jing YANG Shi Xin LUO 《数学学报(英文版)》2005,21(6):1425-1434
Let p be a prime, m ≥ 2, and (m,p(p - 1)) = 1. In this paper, we will calculate explicitly the Gauss sum G(X) = ∑x∈F*qX(x)ζ^Tp^(x) in the case of [(Z/mZ)* : (p)] = 4, and -1 (不属于) (p), where q P^f, f =φ(m)/4, X is a multiplicative character of Fq with order m, and T is the trace map for Fq/Fp. Under the assumptions [(Z/mZ)* : (p)] = 4 and 1(不属于) (p), the decomposition field of p in the cyclotomic field Q(ζm) is an imaginary quartic (abelian) field. And G(X) is an integer in K. We deal with the case where K is cyclic in this oaDer and leave the non-cvclic case to the next paper. 相似文献
15.
The aim of this paper is to study the characters of the maximal subgroup
of the symplectic group Sp
4(q)q-even, where
is the stabilizer of the one-dimensional space <f
1> in Sp
4(q). 相似文献
16.
A Gaussian t-design is defined as a finite set X in the Euclidean space ℝn satisfying the condition:
for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝn, then
. We call X a tight Gaussian 2e-design in ℝn if
holds. In this paper we study tight Gaussian 2e-designs in ℝn. In particular, we classify tight Gaussian 4-designs in ℝn with constant weight
or with weight
. Moreover we classify tight Gaussian 4-designs in ℝn on 2 concentric spheres (with arbitrary weight functions). 相似文献
17.
In this paper we classify point sets of minimum size of two types (1) point sets meeting all secants to an irreducible conic
of the desarguesian projective plane PG(2,q), q odd; (2) point sets meeting all external lines and tangents to a given irreducible conic
of the desarguesian projective plane PG(2,q), q even. 相似文献
18.
Singular Integrals and Commutators in Generalized Morrey Spaces 总被引:1,自引:0,他引:1
Lubomiea Softova 《数学学报(英文版)》2006,22(3):757-766
19.
Let
be the Galois ring of characteristic 23 and rank n and let
. We give an explicit construction of Hadamard difference sets in
.}Research supported by NSA grant MDA 904-02-1-0080. 相似文献
20.
Antoine Deza Boris Goldengorin Dmitrii V. Pasechnik 《Journal of Algebraic Combinatorics》2006,23(2):197-203
We show that the symmetry groups of the cut cone Cutn and the metric cone Metn both consist of the isometries induced by the permutations on
, that is,
for n ≥ 5. For n = 4 we have
. This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing.
For instance,
for n ≥ 5, where Hypn denotes the hypermetric cone. 相似文献