Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays ${\mathsf{OA}}_{\lambda }(k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m∕\lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k\ge n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $gcd(m,\lambda )=1$. We construct a basic OA with $n=2$ and $k=4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs with $n=2$, modulo the Hadamard matrix conjecture.     

Constructions of Group Divisible Designs

In this paper we present constructions for group divisible designs from generalized partial difference matrices. We describe some classes of examples.     

New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u ² − u,u ² − u,u ² − u − 1) and 2 − (2u ² + u,u ²,u ² − u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.     

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

All Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291-303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,p^m) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71-87].     

Generating and improving orthogonal designs by using mixed integer programming

Analysts faced with conducting experiments involving quantitative factors have a variety of potential designs in their portfolio. However, in many experimental settings involving discrete-valued factors (particularly if the factors do not all have the same number of levels), none of these designs are suitable.In this paper, we present a mixed integer programming (MIP) method that is suitable for constructing orthogonal designs, or improving existing orthogonal arrays, for experiments involving quantitative factors with limited numbers of levels of interest. Our formulation makes use of a novel linearization of the correlation calculation.The orthogonal designs we construct do not satisfy the definition of an orthogonal array, so we do not advocate their use for qualitative factors. However, they do allow analysts to study, without sacrificing balance or orthogonality, a greater number of quantitative factors than it is possible to do with orthogonal arrays which have the same number of runs.     

Comments on 3‐blocked designs

This note provides a correction and some additions to a 1999 article by Luigia Berardi and Fulvio Zuanni on blocking 3‐sets in designs. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 328–331, 2012     

On finding mixed orthogonal arrays of strength 2 with many 2-level factors

We describe a method for finding mixed orthogonal arrays of strength 2 with a large number of 2-level factors. The method starts with an orthogonal array of strength 2, possibly tight, that contains mostly 2-level factors. By a computer search of this starting array, we attempt to find as large a number of 2-level factors as possible that can be used in a new orthogonal array of strength 2 containing one additional factor at more than two levels. The method produces new orthogonal arrays for some parameters, and matches the best-known arrays for others. It is especially useful for finding arrays with one or two factors at more than two levels.     

Constructions of new orthogonal arrays and covering arrays of strength three

A covering array of size N, strength t, degree k, and order v, or a CA(N;t,k,v) in short, is a k×N array on v symbols. In every t×N subarray, each t-tuple column vector occurs at least once. When ‘at least’ is replaced by ‘exactly’, this defines an orthogonal array, OA(t,k,v). A difference covering array, or a DCA(k,n;v), over an abelian group G of order v is a k×n array (aij) (1?i?k, 1?j?n) with entries from G, such that, for any two distinct rows l and h of D (1?l<h?k), the difference list Δlh={dh1−dl1,dh2−dl2,…,dhn−dln} contains every element of G at least once.Covering arrays have important applications in statistics and computer science, as well as in drug screening. In this paper, we present two constructive methods to obtain orthogonal arrays and covering arrays of strength 3 by using DCAs. As a consequence, it is proved that there are an OA(3,5,v) for any integer v?4 and v?2 (mod 4), and an OA(3,6,v) for any positive integer v satisfying gcd(v,4)≠2 and gcd(v,18)≠3. It is also proved that the size CAN(3,k,v) of a CA(N;3,k,v) cannot exceed v³+v² when k=5 and v≡2 (mod 4), or k=6, v≡2 (mod 4) and gcd(v,18)≠3.     

On the construction of restricted minimum aberration designs

Deriving a formula, we show that an optimal moment design has the property that each treatment combination appears as equally often as possible. Modifying the definition of minimum aberration to restricted minimum aberration, we show that the word length pattern of a design is proportional to that of its complement. Many restricted minimum aberration designs are constructed.     

On affine designs and Hadamard designs with line spreads

Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291-303] described a construction that relates any Hadamard design H on 4^m-1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m,4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m,4) if, and only if, H is the classical design of points and hyperplanes in PG(2m-1,2) and the line spread is of a special type. Computational results about line spreads in PG(5,2) are given. One of the affine designs obtained has the same 2-rank as the design of points and planes in AG(3,4), and provides a counter-example to a conjecture of Hamada [On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973) 153-226].     

构造正交表的一种替换模式

A method of constructing orthogonal arrays is presented by Zhang, Lu and Pang in 1999. In this paper,the method is developed by introducing a replacement scheme on the construction of orthogonal arrays ,and some new mixed-level orthogonal arrays of run size 36 are constructed.     

On quasi‐orthogonal cocycles

We introduce the notion of quasi‐orthogonal cocycle. This is motivated in part by the maximal determinant problem for square ‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi‐Hadamard groups, relative quasi‐difference sets, and certain partially balanced incomplete block designs, are proved.     

A series of Siamese twin designs

A {0,±1}-matrix S is called a Siamese twin design sharing the entries of I if S=I+K−L, where I,K,L are nonzero {0,1}-matrices and both I+K and I+L are incidence matrices of symmetric designs with the same parameters. Let p and 2p+3 be prime powers and . We construct a Siamese twin design with parameters (4(p+1)²,2p²+3p+1,p²+p).     

Some Hadamard designs with parameters (71,35,17)

Up to isomorphisms there are precisely eight symmetric designs with parameters (71, 35, 17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob₂₁ × Z5):Z4. In this case Z(G) = 〈1〉, G′ has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2‐subgroup is elementary abelian. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 144–149, 2002; DOI 10.1002/jcd.996     

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