Simple 3‐designs of PSL(2, 2n) with block size 7

In this paper, we determine the number of the orbits of 7‐subsets of with a fixed orbit length under the action of PSL(2, 2ⁿ). As a consequence, we determine the distribution of λ for which there exists a simple 3‐(2ⁿ + 1, 7, λ) design with PSL(2, 2ⁿ) as an automorphism group. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 1–17, 2008     

On Minimal Defining Sets of Full Designs and Self-Complementary Designs, and a New Algorithm for Finding Defining Sets of t-Designs

A defining set of a t-(v, k, λ) design is a partial design which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper partial designs is a defining set. This paper proposes a new and more efficient algorithm that finds all non-isomorphic minimal defining sets of a given t-design. The complete list of minimal defining sets of 2-(6, 3, 6) designs, 2-(7, 3, 4) designs, the full 2-(7, 3, 5) design, a 2-(10, 4, 4) design, 2-(10, 5, 4) designs, 2-(13, 3, 1) designs, 2-(15, 3, 1) designs, the 2-(25, 5, 1) design, 3-(8, 4, 2) designs, the 3-(12, 6, 2) design, and 3-(16, 8, 3) designs are given to illustrate the efficiency of the algorithm. Also, corrections to the literature are made for the minimal defining sets of four 2-(7, 3, 3) designs, two 2-(6, 3, 4) designs and the 2-(21, 5, 1) design. Moreover, an infinite class of minimal defining sets for 2-((v) || 3){v\choose3} designs, where v ≥ 5, has been constructed which helped to show that the difference between the sizes of the largest and the smallest minimal defining sets of 2-((v) || 3){v\choose3} designs gets arbitrarily large as v → ∞. Some results in the literature for the smallest defining sets of t-designs have been generalized to all minimal defining sets of these designs. We have also shown that all minimal defining sets of t-(2n, n, λ) designs can be constructed from the minimal defining sets of their restrictions when t is odd and all t-(2n, n, λ) designs are self-complementary. This theorem can be applied to 3-(8, 4, 3) designs, 3-(8, 4, 4) designs and the full 3-(8 || 4)3-{8 \choose 4} design using the previous results on minimal defining sets of their restrictions. Furthermore we proved that when n is even all (n − 1)-(2n, n, λ) designs are self-complementary.     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

Large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ have been studied by Schellenberg, Chen, Lindner and Stinson. These large sets have applications in cryptography in the construction of perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for n = 6 and for all n = 3^k, k ≥ 1. In this paper, we present new recursive constructions and use them to show that such large sets exist for all odd n ≡ 0 (mod 3) and for even n = 24m, where m odd ≥ 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 285–296, 2001     

Existence of large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ (denoted by LS (2ⁿ4¹)) were first studied by Schellenberg and Stinson and motivated by their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡ 0 (mod 3) and do exist for any n ? {12, 36, 48, 144} ∪ {m > 6 : m ≡ 6,30 (mod 36)}. In this paper, we show that an LS (2^{12k + 6}4¹) exists for any k ≠ 2. So, the existence of LS (2ⁿ4¹) is almost solved with five possible exceptions n ∈ {12, 30, 36, 48, 144}. This solution is based on the known existence results of S (3, 4, v)s by Hanani and special S (3, {4, 6}, 6m)s by Mills. Partitionable H (q, 2, 3, 3) frames also play an important role together with a special known LS (2¹⁸4¹) with a subdesign LS (2⁶4¹). © 2004 Wiley Periodicals, Inc.     

New large sets of t‐designs with prescribed groups of automorphisms

In this article, we investigate the existence of large sets of 3‐designs of prime sizes with prescribed groups of automorphisms PSL(2,q) and PGL(2,q) for q < 60. We also construct some new interesting large sets by the use of the computer program DISCRETA. The results obtained through these direct methods along with known recursive constructions are combined to prove more extensive theorems on the existence of large sets. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 210–220, 2007     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

We determine the distribution of 3?(q + 1,k,λ) designs, with k ? {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ? 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3?(q + 1,k,λ) designs, with k ? {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.     

Mutually disjoint designs and new 5‐designs derived from groups and codes

The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the table of 5‐designs given in (G. B. Khosrovshahi and R. Laue, t‐Designs with t⩾3, in “Handbook of Combinatorial Designs”, 2nd edn, C. J. Colbourn and J. H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 79–101), namely, 5−(v, k, λ) designs with (v, k, λ)=(18, 8, 2m) (m=6, 9), (19, 9, 7m) (m=6, 9), (24, 9, 6m) (m=3, 4, 5), (25, 9, 30), (25, 10, 24m) (m=4, 5), (26, 10, 126), (30, 12, 440), (32, 6, 3m) (m=2, 3, 4), (33, 7, 84), and (36, 12, 45n) for 2⩽n⩽17. These results imply that a simple 5−(v, k, λ) design with (v, k)=(24, 9), (25, 9), (26, 10), (32, 6), or (33, 7) exists for all admissible values of λ. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 305–317, 2010     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).     

Overlarge sets of disjoint Steiner quadruple systems

In this article, we construct overlarge sets of disjoint S(3, 4, 3ⁿ − 1) and overlarge sets of disjoint S(3, 4, 3ⁿ + 1) for all n ≥ 2. Up to now, the only known infinite sequence of overlarge sets of disjoint S(3, 4, v) were the overlarge sets of disjoint S(3, 4, 2ⁿ) obtained from the oval conics of desarguesian projective planes of order 2ⁿ. © 1999 John Wiley & Sons, Inc. J Combin Design 7: 311–315, 1999     

Single‐cell compact finite‐difference discretization of order two and four for multidimensional triharmonic problems

In this article, we discuss finite‐difference methods of order two and four for the solution of two‐and three‐dimensional triharmonic equations, where the values of u,(?²u/?n²) and (?⁴u/?n⁴) are prescribed on the boundary. For 2D case, we use 9‐ and for 3D case, we use 19‐ uniform grid points and a single computational cell. We introduce new ideas to handle the boundary conditions and do not require to discretize the boundary conditions at the boundary. The Laplacian and the biharmonic of the solution are obtained as byproduct of the methods. The resulting matrix system is solved by using the appropriate block iterative methods. Computational results are provided to demonstrate the fourth‐order accuracy of the proposed methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Bounds on the number of Hadamard designs of even order

A new lower bound on the number of non‐isomorphic Hadamard symmetric designs of even order is proved. The new bound improves the bound on the number of Hadamard designs of order 2n given in [12] by a factor of 8n ? 1 for every odd n > 1, and for every even n such that 4n ? 1 > 7 is a prime. For orders 8, 10, and 12, the number of non‐isomorphic Hadamard designs is shown to be at least 22,478,260, 1.31 × 10¹⁵, and 10²⁷, respectively. For orders 2n = 14, 16, 18 and 20, a lower bound of (4n ? 1)! is proved. It is conjectured that (4n ? 1)! is a lower bound for all orders 2n ≥ 14. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 363‐378, 2001     

Class-uniformly resolvable designs with block sizes 2 and 3

In this article we consider restricted resolvable designs (RRP) with block sizes 2 and 3 that are class uniform. A characterization scheme is developed, based on the ratio a:b of pairs to triples, and necessary conditions are provided for the existence of these designs based on this characterization. We show asymptotic existence results when (a,b) = (1, 2n), n ≥ 1 and when (a,b) = (9, 2). We also study the specific cases when (a,b) = (1, 2n), 1 ≤ n ≤ 5, (a,b) = (3, 6u−2), u ≥ 1 and when (a,b) E {(1, 1), (3, 1), (7, 2), (3, 4), (9, 2)}. © 1996 John Wiley & Sons, Inc.     

PSL(2,q) as a collineation group of projective planes of small order

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q ² with q odd, notably that G acts irreducibly on П for q≠3, 5, 9. The material herein was presented to the University of Toronto in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author is grateful to Professor Chat Y. Ho, presently at the University of Florida, for guidance in this research.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

Competition graphs and clique dimensions

Here it is proved that for almost all simple graphs over n vertices one needs Ω(n^4/3(log n)^?4/3) extra vertices to obtain them as a double competition graph of a digraph. on the other hand O(n^5/3) extra vertices are always sufficient. Several problems remain open.     

Cantor-Bendixson degrees and convexity in ℝ2

We present an ordinal rank, δ³, which refines the standard classification of non-convexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω₁ + 1 when ordered by δ-rank. The rank δ³ (S) of a setS is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a setS is a subset ofS all of whose unordered 3-tuples fail to have their convex hull inS. Similarly, δⁿ (S) is defined for alln>1. The classification cannot be done using δ², which considers only 2-cliques (known in the literature also as “visually independent subsets”), and in dimension 3 or higher the analogous classification is not valid.     

Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.