Arrays for orthogonal designs

A set of square real matrices is said to be amicable if for some permutation σ of the set . An infinite number of arrays which are suitable for any amicable set of eight circulant matrices are introduced. Applications include new classes of orthogonal designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 166–173, 2000     

On the amicability of orthogonal designs

Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2ⁿp, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2ⁿp where p odd, with no zero entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space‐time codes, this paper may have applications in space‐time codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 240‐252, 2009     

Weak amicable T‐matrices and Plotkin arrays

We introduce the class of weak amicable T‐matrices and use it to construct a class of orthogonal designs, for p = 1 and for p a prime power ≡ 3 (mod 4), and all odd q, q ≤ 21. This class includes new Plotkin arrays of order 24, 40, 56 and for the first time, of orders 8q, q ∈ {9,11,13,15,17,19,21}. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 44–52, 2008     

Sequences and arrays involving free variables

Abstact: Sequences in free variables are introduced and used to construct arrays in free variables which are suitable for circulant matrices. Most of the arrays found are maximal in the number of free variables. Applications include many new Goethals‐Seidel type arrays and complex orthogonal designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 17–27, 2001     

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.     

Hadamard matrices of small order and Yang conjecture

We show that 138 odd values of n<10000 for which a Hadamard matrix of order 4n exists have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n=191, 5767, 7081, 8249 in that range for which Hadamard matrices of order 4n exist. There is a unique equivalence class of near‐normal sequences NN(36), and the same is true for NN(38) and NN(40). This means that the Yang conjecture on the existence of near‐normal sequences NN(n) has been verified for all even n⩽40, but it still remains open. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 254–259, 2010     

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     

New Classes of Groups Containing Menon Difference Sets

A theorem due to Davis on the existence of Menon difference sets in 2-groups is generalised to non-2-groups. The existence of Menon difference sets in many new non-abelian groups is established.     

Williamson matrices up to order 59

A recent result of Schmidt has brought Williamson matrices back into the spotlight. In this article, a new algorithm is introduced to search for hard to find Williamson matrices. We find all nonequivalent Williamson matrices of odd order n up to n = 59. It turns out that there are none for n = 35, 47, 53, 59 and it seems that the Turyn class may be the only infinite class of these matrices.      

Enumeration of generalized Hadamard matrices of order 16 and related designs

We investigate signings of symmetric GDD( , 16, )s over for . Beginning with , at each stage of this process a signing of a GDD( , 16, ) produces a GDD( , 16, ). The initial GDDs ( ) correspond to Hadamard matrices of order 16. For , the GDDs are semibiplanes of order 16, and for the GDDs are semiplanes of order 16 which can be extended to projective planes of order 16. In this article, we completely enumerate such signings which include all generalized Hadamard matrices of order 16. We discuss the generation techniques and properties of the designs obtained during the search. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 119–135, 2009     

The D-optimal saturated designs of order 22

This paper attempts to prove the D-optimality of the saturated designs X${}^1$ and X${}^{11}$ of order 22, already existing in the current literature. The corresponding non-equivalent information matrices M${}^1$=(X${}^1$)${}^T$X${}^1$ and M${}^{11}$=(X${}^{11}$)${}^T$X${}^{11}$ have the maximum determinant. Within the application of a specific procedure, all symmetric and positive definite matrices M of order 22 with determinant the square of an integer and $\ge$det(M${}^1$) are constructed. This procedure has indicated that there are 26 such non-equivalent matrices M, for 24 of which the non-existence of designs X such that X${}^T$X =M is proved. The remaining two matrices M are the information matrices M${}^1$ and M${}^{11}$.     

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.     

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     

Constructions for orthogonal designs using signed group orthogonal designs

Craigen introduced and studied signed group Hadamard matrices extensively and eventually provided an asymptotic existence result for Hadamard matrices. Following his lead, Ghaderpour introduced signed group orthogonal designs and showed an asymptotic existence result for orthogonal designs and consequently Hadamard matrices. In this paper, we construct some interesting families of orthogonal designs using signed group orthogonal designs to show the capability of signed group orthogonal designs in generation of different types of orthogonal designs.     

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     

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     

On defining sets of full designs

A defining set of a t-(v,k,λ) design is a subcollection of its blocks which is contained in a unique t-design with the given parameters on a given v-set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M|∣M is a minimal defining set of D}. The unique simple design with parameters is said to be the full design on v elements; it comprises all possible k-tuples on a v set. We provide two new minimal defining set constructions for full designs with block size k≥3. We then provide a generalisation of the second construction which gives defining sets for all k≥3, with minimality satisfied for k=3. This provides a significant improvement of the known spectrum for designs with block size three. We hypothesise that this generalisation produces minimal defining sets for all k≥3.     

Triples of orthogonal Latin and Youden rectangles for small orders

We have performed a complete enumeration of nonisotopic triples of mutually orthogonal Latin rectangles for . Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.