Alternating groups and flag‐transitive 2‐(v,k, 4) symmetric designs

In this article, we study the classification of flag‐transitive, point‐primitive 2‐ (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag‐transitive, point‐primitive nontrivial 2‐ (v, k, 4) symmetric design ?? is an alternating group A_n for n≥5, then (v, k) = (15, 8) and ?? is one of the following: (i) The points of ?? are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A₇ or A₈, and the stabilizer G_x of a point x of ?? is L₃(2) or AGL₃(2), respectively. (ii) The points of ?? are the edges of the complete graph K₆ and the blocks are the complete bipartite subgraphs K_{2, 4} of K₆, G = A₆ or S₆, and G_x = S₄ or S₄ × Z₂, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011     

A classification of flag‐transitive point‐imprimitive 2‐designs with block size 6

We determine all 2‐ designs admitting a flag‐transitive point‐imprimitive automorphism group.     

Flag‐Transitive Point‐Primitive Symmetric (ν,κ,λ) Designs With λ at Most 100

Let be a symmetric (ν,κ,λ) design with λ ≤ 100. If G is a flag‐transitive and point‐primitive automorphism group of , then G must be an affine or almost simple group.     

Flag‐transitive 2‐ designs of product type

In this paper, we prove that if a 2‐ design admits a flag‐transitive automorphism group G, then G is of affine, almost simple type, or product type. Furthermore, we prove that if G is product type then is either a 2‐(25, 4, 12) design or a 2‐(25, 4, 18) design with .     

On Flag‐Transitive Symmetric Designs of Affine Type

It is shown that, if is a nontrivial 2‐ symmetric design, with , admitting a flag‐transitive automorphism group G of affine type, then , p an odd prime, and G is a point‐primitive, block‐primitive subgroup of . Moreover, acts flag‐transitively, point‐primitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.     

Some infinite families of 2‐designs from PG (n,q)

In this paper, we establish the existence of some infinite families of 2‐designs from ‐dimensional projective geometry , which admit ‐dimensional projective special linear group as their flag‐transitive automorphism group.     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

On a question of Thas on partial 3‐(qn + 1, q + 1, 1) designs

In this note, we answer a question of JA Thas about partial $3$‐ $\left(q^n+1,q+1,1\right)$ designs. We then extend this answer to a result about the embedding of certain partial $3$‐ $\left(q^2+1,q+1,1\right)$ designs into Möbius planes.     

Orthogonal group divisible designs of type hn for h ≤ 6

For the existence problem of OGDDs of type g^u, Colbourn and Gibbons settled it with few possible exceptions for each group size g. In this article, we will completely settle it for g ≤ 6. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Group divisible 3‐designs with block size four and group type 1ns1

In this article, we mainly consider the existence problem of a group divisible design GDD $\left(3,4,n+s\right)$ of type $1^n{s}^1$. We present two recursive constructions for this configuration using candelabra systems and construct explicitly a few small examples admitting given automorphism groups. As an application, several new infinite classes of GDD $\left(3,4,n+s\right)$s of type $1^n{s}^1$ are produced. Meanwhile a few new infinite families on candelabra quadruple systems with group sizes being odd and stem size greater than one are also obtained.     

The spectrum of cyclic BSA(ν, 3, λ; α) with α = 2, 3

Balanced sampling plans excluding contiguous units (or BSEC) were first introduced by Hedayat, Rao, and Stufken in 1988. In this paper, we generalize the concept of a cyclic BSEC to a cyclic balanced sampling plan to avoid the selection of adjacent units (or CBSA for short) and use Langford and extended Langford sequences to construct a cyclic BSA(ν, 3, λ; α) with α = 2, 3. We finally establish the necessary and sufficient conditions for the existence of a cyclic BSA(ν, 3, λ; α) where α = 2, 3. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Infinite families of 3‐designs from APN functions

Combinatorial $t$‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing $t$‐designs is by the action of a permutation group that is $t$‐transitive or $t$‐homogeneous on a point set. This approach produces $t$‐designs, but may not yield $\left(t+1\right)$‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented.     

A class of group divisible 3‐designs and their applications

In this article, we first show that a group divisible 3‐design with block sizes from {4, 6}, index unity and group‐type 2^m exists for every integer m≥ 4 with the exception of m = 5. Such group divisible 3‐designs play an important role in our subsequent complete solution to the existence problem for directed H‐designs DH_λ(m, r, 4, 3)s. We also consider a way to construct optimal codes capable of correcting one deletion or insertion using the directed H‐designs. In this way, the optimal single‐deletion/insertion‐correcting codes of length 4 can be constructed for all even alphabet sizes. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 136–146, 2009     

α‐Resolvable group divisible designs with block size three

A group divisible design GD(k,λ,t;tu) is α‐resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(u ? 1) = r(k ? 1), bk = rtu, k|αtu and α|r. It is shown in this paper that these conditions are also sufficient when k = 3, with some definite exceptions. © 2004 Wiley Periodicals, Inc.     

Exceptions and characterization results for type‐1 λ‐designs

Let $X$ be a finite set with $v$ elements, called points and $\beta$ be a family of subsets of $X$, called blocks. A pair $\left(X,\beta \right)$ is called $\lambda$‐design whenever $\mid \beta \mid =\mid X\mid$ and

• 1. for all $B_i,B_j\in \beta ,i\ne j,\mid B_i\cap B_j\mid =\lambda$;
• 2. for all $B_j\in \beta ,\mid B_j\mid =k_j>\lambda$, and not all $k_j$ are equal.

The only known examples of $\lambda$‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$‐designs are type‐1. Let $r,r^{*}?(r>r^{*})$ be replication numbers of a $\lambda$‐design $D=\left(X,\beta \right)$ and $g=\text{gcd}\left(r?1,r^{*}?1\right),m=\text{gcd}\left(\left(r?r^{*}\right)∕g,\lambda \right)$, and $m\prime =m$, if $m$ is odd and $m\prime =m∕2$, otherwise. For distinct points $x$ and $y$ of $D$, let $\lambda \left(x,y\right)$ denote the number of blocks of $X$ containing $x$ and $y$. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if $\frac{r\left(r?1\right)}{\left(v?1\right)}?\frac{k\left(r?r^{*}\right)}{m\prime \left(v?1\right)}$ are not integers for $k=1,2,\text{…},m\prime ?1$, then $D$ is type‐1. As an application of these results, we show that for fixed positive integer $\theta$ there are finitely many nontype‐1 $\lambda$‐designs with $r=r^{*}+\theta$. If $r?r^{*}=27$ or $r?r^{*}=4p$ and $r^{*}\ne {\left(p?1\right)}^2$, or $v=7p+1$ such that $p?1,13\left(\mathrm{mod}21\right)$ and $p?4,9,19,24\left(\mathrm{mod}35\right)$, where $p$ is a positive prime, then $D$ is type‐1. We further obtain several inequalities involving $\lambda (x,y)$, where equality holds if and only if D is type‐1.     

On the existence of pure (v, 4, λ)-PMD with λ = 1 and 2

In this article, we investigate the existence of pure (v, 4, λ)-PMD with λ = 1 and 2, and obtain the following results: (1) a pure (v, 4, 1)-PMD exists for every positive integer v = 0 or 1 (mod 4) with the exception of v = 4 and 8 and the possible exception of v = 12; (2) a pure (v, 4, 2)-PMD exists for every integer v ≥ 6. © 1994 John Wiley & Sons, Inc.