Non-symmetric 2-designs admitting a two-dimensional projective linear group

This paper is a contribution to the study of non-symmetric 2-designs admitting a flag-transitive automorphism group. We prove that if \(\mathcal {D}\) is a non-trivial non-symmetric 2-\((v, k, \lambda )\) design with \((r,\lambda ) = 1\) and \(G=PSL(2,q)\) acts flag-transitively on \(\mathcal {D}\), then up to isomorphism \(\mathcal {D}\) is a unique Witt-Bose-Shrikhande space, a unique 2-(6, 3, 2) design, a unique 2-(8, 4, 3) design, a unique 2-(10, 6, 5) design, or a unique 2-(28, 7, 2) design.     

On 2-designs

Denote by M_v the set of integers b for which there exists a 2-design (linear space) with v points and b lines. M_v is determined as accurately as possible. On one hand, it is shown for v > v₀ that M_v contains the interval [v + p + 1, v + p + q ? 1]. On the other hand for v of the form p² + p + 1 it is shown that the interval [v + 1, v + p ? 1] is disjoint from M_v; and if v > v₀ and p is of the form q² + q, then an additional interval [v + p + 1, v + p + q ? 1] is disjoint from M_v.     

Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

Switching for 2-designs

Designs, Codes and Cryptography - In this paper, we introduce a switching for 2-designs, which defines a type of trade. We illustrate this method by applying it to some symmetric...     

2-designs overGF(2 m )

At – (n, k, ; q) design is a collection ofk-dimensional subspaces of ann-dimensional vector space overGF(q) with the property that anyt-dimensional subspace is contained in exactly members of . It is also called a design over a finite field or aq-analoguet-design. The first nontrivial example fort 2 was given by S. Thomas. Namely, he constructed a series of 2 – (n, 3, 7; 2) design for alln 7 satisfying (n, 6) = 1. Under the same restriction onn, we show that the base field of Thomas' design is extensible toGF(2 ^m ), i.e., we construct a 2 – (n, 3, 2^2m + 2 ^m + 1; 2 ^m ) design for allm 1.Dedicated to Professor Tuyosi Oyama on his 60th Birthday     

Miscellaneous classification results for 2-designs

An orderly algorithm combined with clique searching is used to show that there are—up to isomorphism, in all cases—325,062 resolvable 2-(16,4,2) designs with 339,592 resolutions, 19,072,802 2-(13,6,5) designs, and 15,111,019 2-(14,7,6) designs. Properties of the classified designs are further discussed.     

New 3-designs and 2-designs having U(3,3) as an automorphism group

In this paper we classify $2$-designs and $3$-designs with $28$ or $36$ points admitting a transitive action of the unitary group $U(3,3)$ on points and blocks. We also construct $2$-designs and $3$-designs with $56$ or $63$ points and strongly regular graphs on 36, 63 or 126 vertices having $U(3,3)$ as a transitive automorphism group. Further, we show that this completes the classification of $3$-designs admitting a transitive action of the group $U(3,3)$, in terms of parameters. A number of the $3$-designs and $2$-designs obtained in this paper have not been known before up to our best knowledge.     

Extended (2, 4)-designs

In this paper, the concept of an extended (2, 4)-design is introduced. An extended (2, 4)-design is a pair (X, $\text{B}$) where X is a finite set and $\text{B}$ is a collection of 4-tuples of not necessarily distinct elements of X, such that every pair of not necessarily distinct elements of X is contained in exactly one member of $\text{B}$. It is shown that an extended (2,4)-design of order n exists for every positive integer n except n = 6, 8 and 9. Several inequivalent designs of order n are obtained.     

Strongly uniform extensions of dual 2-designs

A strongly α-uniform partial line space of order (s, t) is called an α-partial geometry. If α = t+1, then the geometry is a dual 2-design. Locally triangular and locally Grassman graphs correspond to triangular extensions of certain dual 2-designs, and the class of strongly uniform quasi-biplanes coincides with the class of strongly uniform extensions of dual 2-designs. We study strongly uniform extensions of dual 2-designs.     

Cyclic resolvability of cyclic Steiner 2-designs

In this article, direct and recursive constructions for a cyclically resolvable cyclic Steiner 2-design are given. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 177–187, 1997     

Quasi-residual 2-designs, 1\tfrac{1}{2}-designs,and strongly regular multigraphs

It is proved that a quasi-residual 2-(v, k, )-design with k>1/2⁵+O(⁴) can be embedded into a symmetric 2-design. This improves a result by Bose, Shrikhande, and Singhi [1]. Our proof uses properties of strongly regular multigraphs and -designs. In particular, we give a simple sufficient condition for a strongly regular multigraph to be isomorphic to the block multigraph of a 2-design.Part of this research was done while the author was at Westfield College, London.     

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.     

On a composition of cyclic 2-designs

A CB(v,k,λ) means a cyclic 2-design of block size k coincidence number λ, and with v points. In this paper, a recursive construction of a CB(v,k,λ) from two or three cyclic 2-designs is given.     

Overpartition pairs modulo powers of 2

An overpartition of $n$ is a non-increasing sequence of positive integers whose sum is $n$ in which the first occurrence of a number may be overlined. In this article, we investigate the arithmetic behavior of $b_k\left(n\right)$ modulo powers of $2$, where $b_k\left(n\right)$ is the number of overpartition $k$-tuples of $n$. Using a combinatorial argument, we determine $b_2\left(n\right)$ modulo $8$. Employing the arithmetic of quadratic forms, we deduce that $b_2\left(n\right)$ is almost always divisible by $2^8$. Finally, with the aid of the theory of modular forms, for a fixed positive integer $j$, we show that $b_{2k}\left(n\right)$ is divisible by $2^j$ for almost all $n$.     

Overpartition function modulo powers of 2

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\),

$$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

     

Broken 2-diamond partitions modulo 5

We consider \(\Delta _2(n)\), the number of broken 2-diamond partitions of n, and give simple proofs of two congruences given by Song Heng Chan.     

On λ-designs

A λ-design is a system of subsets S₁, S₂,…, S_n from an n-set S, n > 3, where |S_i ∩ S_j| = λ for i ≠ j, |S_j| = k_j > λ > 0, and not all k_j, are equal. Ryser [9] and Woodall [101 have shown that each element of S occurs either r₁, or r₂ times (r₁ ≠ r₂) among the sets S₁,…, S_n and r₁ +r₂ = n + 1. Here we: (i) mention most of what is currently known about λ-designs; (ii) provide simpler proofs of some known results; (iii) present several new general theorems; and (iv) apply our theorems and techniques to the calculation of all λ-designs for λ ? 5. In fact, this calculation has been done for all λ ?/ 9 and is available from the author.