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].     

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.     

On construction of strong orthogonal arrays and column-orthogonal strong orthogonal arrays of strength two plus

Computer experiments require space-filling designs with good low-dimensional projection properties.Strong orthogonal arrays are a type of space-filling design that provides better stratifications in low dimensions than ordinary orthogonal arrays. In this paper, we address the problem of constructing strong orthogonal arrays and column-orthogonal strong orthogonal arrays of strength two plus. Existing methods typically rely on regular designs or specific nonregular designs as base orthogonal arrays, limiting the sizes of the final designs.Instead, we propose two general methods that are easy to implement and applicable to a wide range of base orthogonal arrays. These methods produce space-filling designs that can accommodate a large number of factors,provide significant flexibility in terms of run sizes, and possess appealing low-dimensional projection properties.Therefore, these designs are ideal for computer experiments.     

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.     

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.     

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.     

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).     

Division of trinomials by pentanomials and orthogonal arrays

Consider a maximum-length binary shift-register sequence generated by a primitive polynomial f of degree m. Let denote the set of all subintervals of this sequence with length n, where m < n ≤ 2m, together with the zero vector of length n. Munemasa (Finite fields Appl, 4(3): 252–260, 1998) considered the case in which the polynomial f generating the sequence is a trinomial satisfying certain conditions. He proved that, in this case, corresponds to an orthogonal array of strength 2 that has a property very close to being an orthogonal array of strength 3. Munemasa’s result was based on his proof that very few trinomials of degree at most 2m are divisible by the given trinomial f. In this paper, we consider the case in which the sequence is generated by a pentanomial f satisfying certain conditions. Our main result is that no trinomial of degree at most 2m is divisible by the given pentanomial f, provided that f is not in a finite list of exceptions we give. As a corollary, we get that, in this case, corresponds to an orthogonal array of strength 3. This effectively minimizes the skew of the Hamming weight distribution of subsequences in the shift-register sequence. The authors are supported by NSERC of Canada.     

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.     

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.     

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.     

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.