Classification of Flag-Transitive Primitive Symmetric(v,k, λ) Designs with PSL(2, q) as Socle

Let D be a nontrivial symmetric(v, k, λ) design, and G be a subgroup of the full automorphism group of D. In this paper we prove that if G acts flag-transitively, pointprimitively on D and Soc(G) = PSL(2, q), then D has parameters(7, 3, 1),(7, 4, 2),(11, 5, 2),(11, 6, 3) or(15, 8, 4).     

Balanced Incomplete Block Designs with Block Size 9 and λ=2,4,8

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: $$\begin{gathered} {\text{ }}\lambda (v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \lambda v(v - 1) \equiv 0{\text{ mod (}}k - 1{\text{)}} \hfill \\ \end{gathered} $$ In this paper we study k=9 with λ=2,4 or 8. For λ=8, we show these conditions on v are sufficient, and for λ=2, 4 respectively there are 8 and 3 possible exceptions the largest of which are v=1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs.     

The spectrum of cyclic (3, λ)-GDD of type g~v

In this paper, it is shown that the necessary conditions for the existence of a ( gv, {g, 3 α }, 3, λ)-DF in Z gv for α∈ {0, 1, 2} are also sufficient with two exceptions of (v, g, λ, α) = (9, 1, 1, 1), (9, 1, 2, 2). Finally, the existence spectrum of a cyclic (3, λ)-GDD of type g v is determined.     

The spectrum of (gv, g, 3, λ)-DF in Z_(gv)

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)- difference family in Zgv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type gv without short orbits is determined.     

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

THE NEAR DIFFERENCE SETS OF TYPE 2 WITH PARAMETERS (9λ+6, 3λ+1, λ)

It is proved in this paper that the number of the near difference sets of type 2 with parameters ( 9λ +6, 3λ+1, λ) is finite by studying the relationship between the cyclic sequence and its normai subsequences. It is determined accordingly that there exists a ( 9λ + 6, 3λ + 1, λ) -near difference set of type 2 if and only if λ. = 1 or 2. Together with the results of [1,2,3,4] we can conclude that there are only 6 groups of parameters corresponding to the near difference sets of type 2 if k ≤ 3λ 4+2 .     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

The λ- structure of the green ring Of GL(2,F p) in characteristic,IV

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F _p), and Sl(2,F _p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F _p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.     

A new class of symmetric (v,k, λ)-designs

We construct new families of symmetric (v, k, )-designs with parameters

Classification of spreads ofPG(3, 4)∖PG(3, 2)

We show that the 800 spreads ofPG(3, 4)PG(3, 2) fall into three orbits of sizes 120, 120 and 560, under the action of its automorphism group.     

The existence of (2 × c, λ) grid-block designs with {c\in \{3, 4, 5\}} and λ ≥ 1

Let X be a v-set and ${\mathcal{B}}$ a collection of r × c arrays with elements in X. Two elements of X are collinear if they are on the same grid line (row or column). A pair ${(X, \mathcal{B})}$ is called an (r × c, λ) grid-block design if every two distinct elements in X are collinear exactly λ times in the arrays of ${\mathcal{B}}$ . This design has absorbed much attention due to its use in DNA library screening. In this paper, we prove that the necessary conditions for the existence of (2 × c, λ) grid-block designs of order v with ${c\in \{3, 4, 5\}}$ and any integer λ ≥ 1 are also sufficient.     

Application of (v, k, λ)-configuration to generation of a conference key

In order for all participants at video conference to communicate mutually, the conference key should be necessary. In this paper, we present the communication protocol that generates a conference key efficiently based on (v,k,λ)-configuration, one class of block designs, which minimizes message transmission overhead needed for this key. Especially, in the case of λ=1, the protocol requires only O(v√v) messages, where v is the number of participants.     

典型群PSU(3,q~2)与2-(v,k,1)设计

讨论自同构群是酉群 PSU(3 ,q2 ) (q=2 l)的区 -本原的 2 -(v,k,1 )设计 .首先证明了它必是点 -本原的 ,然后确定了这种类型的设计 ,即它只能为 2 -(q3+1 ,q+1 ,1 )设计