On the existence of nested orthogonal arrays

A nested orthogonal array is an OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray. Here r<s and M<N. Necessary conditions for the existence of such arrays are obtained in the form of upper bounds on k, given N, M, s, r and g. Examples are given to show that these bounds are quite powerful in proving nonexistence. The link with incomplete orthogonal arrays is also indicated.     

Construction of nested orthogonal arrays

A (symmetric) nested orthogonal array is a symmetric orthogonal array OA(N,k,s,g) which contains an OA(M,k,r,g) as a subarray, where M<N and r<s. In this communication, some methods of construction of nested symmetric orthogonal arrays are given. Asymmetric nested orthogonal arrays are defined and a few methods of their construction are described.     

Further results on mutually nearly orthogonal Latin squares

Nearly orthogonal Latin squares are useful for conducting experiments eliminating heterogeneity in two directions and using different interventions each at each level. In this paper, some constructions of mutually nearly orthogonal Latin squares are provided. It is proved that there exist 3 MNOLS(2m) if and only if m ≥ 3 and there exist 4 MNOLS(2m) if and only if m ≥ 4 with some possible exceptions.     

Incomplete orthogonal arrays and idempotent orthogonal arrays

First, we shall define idempotent orthogonal arrays and notice that idempotent orthogonal array of strength two are idempotent mutually orthogonal quasi-groups. Then, we shall state some properties of idempotent orthogonal arrays.Next, we shall prove that, starting from an incomplete orthogonal arrayT _EF based onE andF E, from an orthogonal arrayT _G based onG = E – F and from an idempotent orthogonal arrayT _H based onH, we are able to construct an incomplete orthogonal arrayT _(F(G×H))F based onF(G × H) andF. Finally, we shall show the relationship between the construction which lead us to this result and the singular direct product of mutually orthogonal quasi-groups given by Sade [5].     

Further results on the existence of periodic solutions to DDEs with multiple lags

1.IntroductionInthispaperwedealwiththeexistenceofperiodicsolutionstothedifferential-delayequations(DDEs)oftheformwherewesupposeinwhatfollowsthatexist.(itisallowedthatshouldbeincreasing.)WehaveLemmaSuppose(H)holdsandn1isaninteger.Ifthereexistsanintege...     

On incomplete orthogonal arrays

First we prove that, if an incomplete orthogonal array (1, r, s, k, t) does exist, then s ?(r ? t + 1)k. Next, we establish a relation between the existence of incomplete orthogonal arrays and the existence of orthogonal arrays. From this relation, we may bring out the upper bounds of the maximum number of contraints.     

Constructions of augmented orthogonal arrays

Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and AOAs (Discrete Math. 341 (2018), 299–307). In this paper, we show that there is an AOA if and only if there is an OA which can be partitioned into subarrays, each being an OA, and that there is a linear AOA if and only if there is a linear maximum distance separable (MDS) code of length and dimension over , which contains a linear MDS subcode of length and dimension over . Some constructions for AOAs and some new infinite classes of AOAs are also given.     

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 the existence of retransmission permutation arrays

We investigate retransmission permutation arrays (RPAs) that are motivated by applications in overlapping channel transmissions. An RPA is an $n\times n$ array in which each row is a permutation of $\{1,\dots ,n\}$, and for $1?i?n$, all $n$ symbols occur in each $i\times ?\frac{n}{i}?$ rectangle in specified corners of the array. The array has types 1, 2, 3 and 4 if the stated property holds in the top left, top right, bottom left and bottom right corners, respectively. It is called latin if it is a latin square. We show that for all positive integers $n$, there exists a type-1, 2, 3, 4 $\mathrm{RPA}\left(n\right)$ and a type-1, 2 latin $\mathrm{RPA}\left(n\right)$.     

Balanced nested designs and balanced arrays

Balanced nested designs are closely related to other combinatorial structures such as balanced arrays and balanced n-ary designs. In particular, the existence of symmetric balanced nested designs is equivalent to the existence of some balanced arrays. In this paper, various constructions for symmetric balanced nested designs are provided. They are used to determine the spectrum of symmetric balanced nested balanced incomplete block designs with block size 3 and 4.     

Bounds on orthogonal arrays and resilient functions

The only known general bounds on the parameters of orthogonal arrays are those given by Rao in 1947 [J. Roy. Statist. Soc. 9 (1947), 128–139] for general OA_γ(t,k,v) and by Bush [Ann. Math. Stat. 23, (1952), 426–434] [3] in 1952 for the special case γ = 1. We present an algebraic method based on characters of homocyclic groups which yields the Rao bounds, the Bush bound in case t ? v, and more importantly a new explicit bound which for large values of t (the strength of the array) is much better than the Rao bound. In the case of binary orthogonal arrays where all rows are distinct this bound was previously proved by Friedman [Proc. 33rd IEEE Symp. on Foundations of Comput. Sci., (1992), 314–319] in a different setting. We also note an application to resilient functions. © 1995 John Wiley & Sons, inc.     

Further results on the existence,uniqueness and stability of strong solutions of quantum stochastic differential equations

Under a more general Lipschitz condition on the coefficients than our consideration in [E.O. Ayoola, Existence and stability results for strong solutions of quantum stochastic differential equations, Stochastic Anal. Appl. 20 (2) (2002) 263–281], we establish the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations (QSDE). This enables us to exhibit a class of Lipschitzian QSDE whose coefficients are continuous on the locally convex space of solution.     

Multimagic rectangles based on large sets of orthogonal arrays

Large sets of orthogonal arrays (LOAs) have been used to construct resilient functions and zigzag functions by Stinson. In this paper, an application of LOAs in constructing multimagic rectangles is given. Further, some recursive constructions for multimagic rectangles are presented, and some infinite families of bimagic rectangles are obtained.     

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.     

Sub-latin squares and incomplete orthogonal arrays

The main result of this paper is that for any pair of orthogonal Latin squares of side k, there will exist for all sufficiently large n a pair of orthogonal Latin squares with the first pair as orthogonal sub-squares. The orthogonal array corresponding to a set of pairwise orthogonal Latin squares, minus the sub-array corresponding to orthogonal sub-squares is called an incomplete orthogonal array; this concept is generalized slightly.     

Further results on the Aanderaa-Rosenberg conjecture

Some results on the “evasiveness” of graph properties are obtained, extending the work of Rivest and Vuillemin. In particular, it is shown that any nontrivial monotone graph property on n vertices is at least $\text{n}{}^2\text{9}\text{-}\text{evasive}$; particular stronger results are obtained for values of n that are powers of primes less than 14. A new result allowing interpolation among n values is obtained, as are some stronger results on restricted classes of properties.