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1.
The dimension of a combinatorial design ${{\mathcal D}}$ over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of ${{\mathcal D}}$ as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design ${{\mathcal D}}$ that has the same parameters as the complement of a classical point-hyperplane design PG n-1(n, q) or AG n-1(n, q) is greater than or equal to n + 1, with equality if and only if ${{\mathcal D}}$ is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PG d (n, q), where 1 ≤ d ≤ n ? 1, in terms of associated codes defined over some extension field E?=?GF(q t ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AG d (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG d (n, q) for d = 1 and for d > (n ? 2)/2. 相似文献
2.
The nonintersecting classes ? p,q are defined, with p, q ?? ? and p ?? q ?? 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ?? ? p,q , then the complexity c(M) and the Euler characteristic ??(M) of M are related by the formula c(M) = p???(M). The classes ? q,q , q ?? 1, and ?2,1 are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ?3,1 and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ?-invariants of manifolds. 相似文献
3.
Sean McGuinness 《Annals of Combinatorics》2012,16(1):107-119
Erdős and Gallai showed that for any simple graph with n vertices and circumference c it holds that
| E(G) | £ \frac12(n - 1)c{{{\mid}{E(G)}{\mid} \leq {\frac{1}{2}}(n - 1)c}}. We extend this theorem to simple binary matroids having no F
7-minor by showing that for such a matroid M with circumference c(M) ≥ 3 it holds that
| E(M) | £ \frac12r(M)c(M){{{\mid}{E(M)}{\mid} \leq {\frac{1}{2}}r(M)c(M)}}. 相似文献
4.
In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111(n, q; K). Firstly we get the complete Einstein Kahler metric with explicit form on Y111(n, q; K) in the case of K=q/2 + 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111(n, q; K) in case of K=q/2+1/q-1. 相似文献
5.
E.L. Monte Carmelo 《Discrete Mathematics》2008,308(17):3986-3991
Some classes of configurations in projective planes with polarity are constructed. As the main result, lower bounds for the Ramsey numbers r(n)=r(C4;K1,n) are derived from these geometric structures, which improve some bounds due to Parsons about 30 years ago, and also yield a new class of optimal values: r(q2-2q+1)=q2-q+1 whenever q is a power of 2. Moreover, the constructions also imply a known result on C4-K1,n bipartite Ramsey numbers. 相似文献
6.
V. A. Belonogov 《Proceedings of the Steklov Institute of Mathematics》2013,283(1):6-23
Previously, the author made the following conjecture: if a finite group has two semiproportional irreducible characters φ and ψ, then φ(1) = ψ(1). In the present paper, a new confirmation of the conjecture is obtained. Namely, the conjecture is verified for symplectic groups Sp4(q) and PSp4(q). 相似文献
7.
Let GF(q) be a finite field of q elements. Let G denote the group of matrices M(x, y) = (y x0 1) over GF(q) with y ≠ 0. Fix an irreducible polynomial For each a ϵ GF(q), let Xa be the graph whose vertices are the q2 − q elements of G, with two vertices M(x, y), M(v, w) joined by an edge if and only if The graphs Xa with a ϵ/ {0, t2 − 4n} are (q + 1)-regular connected graphs which have received recent attention, as they've been shown to be Ramanujan graphs. We determine the diameter of these graphs Xa. © 1996 John Wiley & Sons, Inc. 相似文献
8.
Every elliptic quartic Γ4 of PG(3,q) with nGF(q)-rational points provides a near-MDS code C of length n and dimension 4 such that the collineation group of Γ4 is isomorphic to the automorphism group of C. In this paper we assume that GF(q) has characteristic p>3. We classify the linear collineation groups of PG(3,q) which can preserve an elliptic quartic of PG(3,q). Also, we prove for q?113 that if the j-invariant of Γ4 does not disappear, then C cannot be extended in a natural way by adding a point of PG(3,q) to Γ4. 相似文献
9.
Charles Semple 《Graphs and Combinatorics》1999,15(4):441-462
Let k be an integer exceeding one. The class of k-regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank-r regular matroid has the maximum number of points if and only if it is isomorphic to M(K
r+1), the cycle matroid of the complete graph on r+1 vertices. A simple rank-r near-regular matroid has the maximum number of points if and only if it is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3-point line of M(K
r+2) and then contracting this point. This paper determines the maximum number of points that a simple rank-r k-regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such
matroid. This matroid is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(K
r+k+1) isomorphic to M(K
k+2) and then contracting each of these points.
Revised: July 27, 1998 相似文献
10.
Vladimir D. Tonchev 《Designs, Codes and Cryptography》2003,29(1-3):247-250
A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)-incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d q of a design D over GF(q) is defined as the minimum value of the q-rank of a generalized incidence matrix of D. It is proved that the dimension d q of the complete design on n points having as blocks all w-subsets, is greater that or equal to n ? w + 1, and the equality d q = n ? w + 1 holds if and only if there exists an [n, n ? w + 1, w] MDS code over GF(q), or equivalently, an n-arc in PG(w ? 2, q). 相似文献
11.
In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E 7(q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E 7(q)) is necessarily isomorphic to E 7(q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively. 相似文献
12.
Allan Lo 《Combinatorica》2016,36(4):471-492
Let K c n be an edge-coloured complete graph on n vertices. Let Δmon(Kc n) denote the largest number of edges of the same colour incident with a vertex of Kc n. A properly coloured cycleis a cycle such that no two adjacent edges have the same colour. In 1976, BollobÁs and Erd?s[6] conjectured that every Kc n with Δmon(Kc n)<?n/2?contains a properly coloured Hamiltonian cycle. In this paper, we show that for any ε>0, there exists an integer n0 such that every Kc n with Δmon(Kc n)<(1/2–ε)n and n≥n0 contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin [1]. Hence, the conjecture of BollobÁs and Erd?s is true asymptotically. 相似文献
13.
Eiichi Bannai Sung Yell Song Hiromichi Yamada 《Journal of Applied Mathematics and Computing》2008,26(1-2):125-131
The character tables of the commutative association schemes coming from the action of the Chevalley group G 2(q) on the set Ω ε of hyperplanes of type O 6 ε (q) in the seven dimensional orthogonal geometry over GF(q) related to the orthogonal group O 7(q) are constructed by modifying the character tables of the association schemes obtained from the action of O 7(q) on Ω ε . 相似文献
14.
《Journal of Combinatorial Theory, Series A》2002,97(1):85-107
A generalization of the Pless symmetry codes to different fields is presented. In particular new infinite families of self-dual codes over GF(4), GF(5), GF(7), and GF(9) are introduced. It is proven that the automorphism group of some of these codes contains the group PSL2(q). New codes over GF(4) and GF(5), with better minimum weight than previously known codes, are given. 相似文献
15.
Clément de Seguins Pazzis 《Linear algebra and its applications》2011,435(11):2708-2721
Let (K) be a field. Given an arbitrary linear subspace V of Mn(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra Mn(K). Here, we strengthen this statement in three ways: we show that Mn(K) is spanned by the products of the form AB with (A,B)∈V2; we prove that every matrix in Mn(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of Mn(K) and n>2, we show that every matrix in Mn(K) is a product of two elements of V. 相似文献
16.
Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→xq. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k. 相似文献
17.
《Journal of Combinatorial Theory, Series B》1986,41(2):250-252
The purpose of this note is to present a counterexample to a conjecture of Kahn and Seymour on the minor-minimal matroids not representable over GF(4). 相似文献
18.
It was proved that the complexity of square root computation in the Galois field GF(3s), s = 2kr, is equal to O(M(2k)M(r)k + M(r) log2r) + 2kkr1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3s) is equal to O(M(2k)M(r)) and O(M(2k)M(r)) + r1+o(1), respectively. If the basis in the field GF(3r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2kkr1+o(1) and r1+o(1) can be omitted. For M(n) one may take n log2nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form Or (M (s) log 2s) = s (log 2s)2ψ(s). 相似文献
19.
Robert E Bixby 《Journal of Combinatorial Theory, Series B》1979,26(2):174-204
In this paper we prove a stronger version of a result of Ralph Reid characterizing the ternary matroids (i.e., the matroids representable over the field of 3 elements, GF(3)). In particular, we prove that a matroid is ternary if it has no seriesminor of type Ln for n ≥ 5 (n cells and n circuits, each of size n ? 1), and no series-minor of type (dual of L5), BII (Fano matroid) or BI (dual of type BII). The proof we give does not assume Reid's theorem. Rather we give a direct proof based on the methods (notably the homotopy theorem) developed by Tutte for proving his characterization of regular matroids. Indeed, the steps involved in our proof closely parallel Tutte's proof, but carrying out these steps now becomes much more complicated. 相似文献
20.
Given an m×n matrix M over E=GF(qt) and an ordered basis A={z1,…,zt} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M1 with entries from the field K. Let r=rank(M) and r1=rank(M1). We show that r?r1?min{rt,n}, and we determine the number b(m,n,r,r1,q,t) of m×n matrices M of rank r over GF(qt) with associated mt×n matrix M1 of rank r1 over GF (q). 相似文献