Existence of (2, 8) GWhD( v) and (4, 8) GWhD( v) with $${ v \equi v 0,1 (mod 8)}$$

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if . We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.      

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

Minimal Blocking Sets in PG(2, 8) and Maximal Partial Spreads in PG(3, 8)

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q ²–q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.     

Embeddings of Sz(32) in E8(5)

We show that the Suzuki group Sz(32) is a subgroup of E₈(5),and so is its automorphism group. Both are unique up to conjugacyin E₈(F) for any field F of characteristic 5, and the automorphismgroup Sz(32):5 is maximal in E₈(5). 1991 Mathematics SubjectClassification 20E28.     

Two maximal subgroups of E8(2)

We show that E₈(2) has a unique conjugacy class of subgroups isomorphic to PSp₄(5) and a unique conjugacy class of subgroups isomorphic to PSL₃(5). There normalizers are maximal subgroups of E₈(2) and are, respectively, isomorphic to PGSp₄(5) and Aut(PSL₃(5)).     

试题研讨(8)

题1(2002年上海市春季高考题)如图1,若从点O所作的两条射线OM、ON上分别有点M1、M2与点N1、N2,则三角形面积之比S△OM1N1/S△OM2N2=OM1/OM2·ON1/ON2.如图2,若从点O所作的不在同一平面内的三条射线OP、OQ和OR上,分别有点P1、P2,点Q1、Q2和点R1、R2,则类似的结论为_______.     

不动点集为RP(8)(U) P(8,2n-1)的对合

设(M,T)是一个带有光滑对合T的光滑闭流形,T的不动点集为RP(8)() P(8,2n-1).本文证明了(M,T)必协边于(RP(8)×RP(8),twist)和(P(8,RP(2n)),T')之一.     

Eine charakterisierung des steinersystems S(5, 8, 24)

S(5, 8, 24) is characterized as the unique Steiner system S(t, k, n) satisfying n = (t + 1)(k ? t + 1) and k ? t + 2 ? 4.     

The value of the Ramsey number r(3, 8)

The Ramsey number R(3, 8) can be defined as the least number n such that every graph on n vertices contains either a triangle or an independent set of size 8. With the help of a substantial amount of computation, we prove that R(3, 8)=28.     

Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)

Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate its behaviour, in particular in the case of the 2-element field. We show that it is compatible with direct products of forms and how it is related to its values on dimension n ? 1. Moreover, it turns out that it is full up to dimension 7. On the other hand, on higher dimensions it is no more full and it is necessary to generalize it to obtain (using computer) a classification of forms on dimension 8 over the 2-element field. This classification is provided, together with the sizes of stabilizers of the corresponding forms.     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

On the Ramsey numbers R(3, 8) and R(3, 9)

Using methods developed by Graver and Yackel, and various computer algorithms, we show that 28 ≤ R(3, 8) ≤ 29, and R(3, 9) = 36, where R(k, l) is the classical Ramsey number for 2-coloring the edges of a complete graph.     

Every 8-uniform 8-regular hypergraph is 2-colorable

As is well known, Lovász Local Lemma implies that everyd-uniformd-regular hypergraph is 2-colorable, providedd 9. We present a different proof of a slightly stronger result; everyd-uniformd-regular hypergraph is 2-colorable, providedd 8.Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation.     

Subsemilattice of a Semilattice of Class Σ2(X, 8)

We give a complete classification of all XI-subsemilattices of the semilattice of the class Σ₂(X, 8). We derive formulas for calculating the numbers of semilattices of the given class.     

Some resolutions of S(5, 8, 24)

There exist 13 mutually disjoint resolutions of the Steiner system S(5, 8, 24). There also exist nine nonisomorphic mutually disjoint resolutions of S(5, 8, 24) where three of the resolutions have the same L₂(23) as an automorphism group and the other six have the same affine group C₂₃¹¹ as an automorphism group. A resolution of S(5, 8, 24) using a group of order 21 is displayed and a 13-dimensional Room-type design is mentioned.