New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Some results on the existence of large sets of t‐designs

A set of trivial necessary conditions for the existence of a large set of t‐designs, LS[N](t,k,ν), is for i = 0,…,t. There are two conjectures due to Hartman and Khosrovshahi which state that the trivial necessary conditions are sufficient in the cases N = 2 and 3, respectively. Ajoodani‐Namini has established the truth of Hartman's conjecture for t = 2. Apart from this celebrated result, we know the correctness of the conjectures for a few small values of k, when N = 2 and t ≤ 6, and also when N = 3 and t ≤ 4. In this article, we show that similar results can be obtained for infinitely many values of k. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 144–151, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10027     

Plateaued functions,partial geometric difference sets,and partial geometric designs

Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ${\mathbb{Z}}_2$‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.     

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     

On optimal third order rotatable designs

We obtain results for choosing optimal third order rotatable designs for the fitting of a third order polynomial response surface model, for m3 factors. By representing the surface in terms of Kronecker algebra, it can be established that the two parameter family of boundary nucleus designs forms a complete class, under the Loewner matrix ordering. In this paper, we first narrow the class further to a smaller complete class, under the componentwise eigenvalue ordering. We then calculate specific optimal designs under Kiefer's % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaadchaaeqaaaaa!38C6!\[\phi _p \] (which include the often used E-, A-, and D-criteria). The E-optimal design attains a particularly simple, explicit form.N. R. D. is grateful for the partial support from the Scientific and Environmental Affairs Division of the North Atlantic Treaty Organization, and from the National Security Agency through Grant MDA904-95-H-1020.     

New orthogonal designs of order 56

Amicable sets of eight matrices are very useful in the construction of orthogonal designs using the Kharaghani array. In this article we use a simple procedure to construct many new amicable sets of eight matrices of order 7 and then new orthogonal designs of order 56. Some of these are restricted to be short amicable sets of two or four circulant matrices.. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 387–393, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10030     

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1) RBIBDs containing a (w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v?4w and . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever and whenever w?1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v,{K,w^*},1)-PBD.If the list of permitted block sizes, K₅, contains all integers of size 5 or more, and v,w∈K₅, then a necessary condition on this PBD is v?4w+1. We show this condition is not sufficient for any w?5 and give the complete spectrum (in v) for 5?w?8, as well as showing the condition v?5w is sufficient with some definite exceptions for w=5 and 6, and some possible exceptions when w=15, namely 77?v?79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4).If the list of permitted block sizes, K₁₍₄₎, contains all integers , and v,w∈K₁₍₄₎, then a necessary condition on this PBD is v?4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w?1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1).     

A few more large sets of t-designs

We construct several new large sets of t-designs that are invariant under Frobenius groups, and discuss their consequences. These large sets give rise to further new large sets by means of known recursive constructions including an infinite family of large sets of 3 − (v, 4, λ) designs. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 293–308, 1998     

On the existence of mandatory representation designs with block sizes 3 and k

A mandatory representation design MR[ν,K] is a pairwise balanced design on ν points with block sizes from K in which for each k ∈ K there is a block in the design of size k. Mendelsohn and Rees [4] investigated the existence of MR[ν,K]s, where 3 ∈ K. In this report we consider additional necessary conditions, where K = {3,k}. These conditions are proved to be sufficient for 4 ≤ k ≤ 50 with one genuine exception. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 122–131, 2000     

On Steinhaus sets

We give a common proof of several results on Steinhaus sets in for d2 including the fact that a Steinhaus set in must be disconnected.     

On simple and supersimple transversal designs

In this paper, the existence of a transversal design TD_λ (4, g) is proved for all indices λ satisfying 2 ≤ λ ≤ g such that any two of its blocks intersect in at most two elements. Similar results are obtained for transversal designs without repeated blocks. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 311–320, 2000     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

On blocking sets of inversive planes

Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild 1 that |S| ≥ 2q for q ≥ 9. We prove that if q is sufficiently large, C is a fixed natural number and |S = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q ≤ 5 and the sizes of some examples of minimal blocking sets in planes of order q ≤ 37 are given. Geometric properties of some of these blocking sets are also studied. © 2004 Wiley Periodicals, Inc.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

A weak difference set construction for higher dimensional designs

We continue the analysis of de Launey's modification of development of designs modulo a finite groupH by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result.We extend the characterization of allAEFs from the cyclic group case to the case whereH is an arbitrary finite abelian group.We prove that ourn-dimensional designs have the form (f(j ₁ j ₂ ...j _n )) (j _i J), whereJ is a subset of cardinality |H| of an extension groupE ofH. We say these designs have a weak difference set construction.We show that two well-known constructions for orthogonal designs fit this development scheme and hence exhibit families of such Hadamard matrices, weighing matrices and orthogonal designs of orderv for which |E|=2v. In particular, we construct proper higher dimensional Hadamard matrices for all orders 4t100, and conference matrices of orderq+1 whereq is an odd prime power. We conjecture that such Hadamard matrices exist for all ordersv0 mod 4.     

Structure and automorphism groups of Hadamard designs

Let n be the order of a Hadamard design, and G any finite group. Then there exists many non-isomorphic Hadamard designs of order 2^{12|G| + 13} n with automorphism group isomorphic to G.This research was supported in part by the National Science Foundation.