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].     

Bounds for the variance of functions of random variables by orthogonal polynomials and Bhattacharya bounds

Upper and lower bounds for the variance of a function g of a random variable X are obtained by expanding g in a series of orthogonal polynomials associated with the distribution of X or by using the convergence of Bhattacharya bounds for exponential families of distribution.     

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.     

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.     

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.     

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.     

Bounds for Bessel functions

We establish lower and upper bounds for the Bessel functionJ _v (x) and the modified Bessel functionI _v(x) of the first kind. Our chief tool is the differential equation satisfied by these functions.     

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.     

Ideal ramp schemes and augmented orthogonal arrays

Ramp schemes were invented in 1985 by C.R. Blakley and C. Meadows. An $(s,t,n)$-ramp scheme is a generalization of a threshold scheme in which there are two thresholds. Recently, D.R. Stinson established the equivalence of ideal ramp schemes and augmented orthogonal arrays. In this study, some new construction methods for augmented orthogonal arrays are presented and then some new augmented orthogonal arrays are obtained; furthermore, we also provide parameter situations where ideal ramp schemes exist for these obtained augmented orthogonal arrays.     

Three characterizations of non-binary correlation-immune and resilient functions

A functionf(X ₁,X ₂, ...,X _n ) is said to betth-order correlation-immune if the random variableZ=f(X ₁,X ₂,...,X _n ) is independent of every set oft random variables chosen from the independent equiprobable random variablesX ₁,X ₂,...,X _n . Additionally, if all possible outputs are equally likely, thenf is called at-resilient function. In this paper, we provide three different characterizations oft th-order correlation immune functions and resilient functions where the random variable is overGF (q). The first is in terms of the structure of a certain associated matrix. The second characterization involves Fourier transforms. The third characterization establishes the equivalence of resilient functions and large sets of orthogonal arrays.     

Quadrature and orthogonal rational functions

Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence.     

Bounds for modified bessel functions

Two-side inequalities for the modified Bessel functionI _v(x), K_v(x) of the first and third kind and of order v, are established. The chief tool is the monotonocity of the functionsI _v+1(x)/I _v(x),K _v+1(x)/K _v(x).     

Generalized latin matrix and construction of orthogonal arrays

In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.     

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.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Orthogonal F-rectangles,orthogonal arrays,and codes

The relationships between a set of orthogonal F-squares or F-rectangles and orthogonal arrays are described. The relationship between orthogonal arrays and error-correcting codes is demonstrated. The development of complete sets of orthogonal F-rectangles allows construction of codes of any word length and for any number of words. Likewise, the development of F-rectangle theory makes code construction much more flexible in terms of a variable number of symbols. The relationship among sets of orthogonal hyperrectangles, orthogonal arrays, and codes is also described.     

Generalized orthogonal arrays: Constructions and related graphs

Generalized orthogonal arrays were first defined to provide a combinatorial characterization of (t, m, s)-nets. In this article we describe three new constructions for generalized orthogonal arrays. Two of these constructions are straightforward generalizations of constructions for orthogonal arrays and one employs new techniques. We construct families of generalized orthogonal arrays using orthogonal arrays and provide net parameters obtained from our constructions. In addition, we define a set of graphs associated with a generalized orthogonal array which provide information about the structure of the array. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 31–39, 1999