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We describe the computer-aided classification of equitable partitions of the 12-cube with quotient matrix [[2, 10], [6, 6]], or, equivalently, simple orthogonal arrays OA(1536, 12, 2, 7), or order-7 correlation-immune Boolean functions in 12 arguments with 1536 ones (which completes the classification of unbalanced order-7 correlation-immune Boolean functions in 12 arguments and, as derived objects, unbalanced order-6 correlation-immune Boolean functions in 11 arguments). We find that there are 103 equivalence classes of the considered objects, and there are only two almost-OA(1536, 12, 2, 8) among them. Additionally, we find that there are 40 equivalence classes of pairs of disjoint simple OA(1536, 12, 2, 7) (equivalently, equitable partitions of the 12-cube with quotient matrix [[2, 6, 4], [6, 2, 4], [6, 6, 0]]) and discuss the existence of a non-simple OA(1536, 12, 2, 7). [ABSTRACT FROM AUTHOR]

Strong orthogonal arrays (SOAs) have better space-filling properties than ordinary orthogonal arrays for computer experiments. SOAs of strengths two plus and two star can improve the space-filling properties in low dimensions, and column orthogonality plays a vital role in computer experiments. In this paper, we use matchings of the bipartite graphs and Hall's theorem to present several constructions of column-orthogonal SOAs of strengths two plus and two star. We also give a new direct construction of SOAs of strength two plus. Our constructions can provide larger numbers of factors of SOAs and column-orthogonal SOAs of strengths two plus and two star than those in the existing literature, having flexible run sizes. The constructions are convenient and flexible, and the resulting designs are good for computer experiments. [ABSTRACT FROM AUTHOR]

This work brings together several types of combinatorial designs: difference matrices, difference covering arrays and difference schemes by defining the concept of covering scheme of strength t over an abelian additive group. Connections of covering schemes with orthogonal arrays and covering arrays are also established. We show general results of covering schemes of strength t using a method based on the factorization of a group and some refinements for particular classes. We apply the previous results to investigate covering schemes having three, four and five factors. Finally, a reformulation of covering schemes in terms of graph theory is established. [ABSTRACT FROM AUTHOR]

Orthogonal array is one of the important research subjects in combinatorial design theory and experimental design theory, and it is widely applied to statistics, computer science, coding theory and cryptography. There are many constructions and results for orthogonal array of strength 2, however the results on orthogonal array of strength t ≥ 3 are rare. In this paper, we first present two new effective constructions for orthogonal arrays of strength t ≥ 3 based on large sets of orthogonal arrays. Second, many infinite families of large sets of orthogonal arrays are obtained and then some new series of orthogonal array of strength t ≥ 3 are produced. [ABSTRACT FROM AUTHOR]

The concept of detecting arrays was developed to locate and detect interaction faults arising between the factors in a component-based system during software testing. In this paper, we propose a family of consecutive detecting arrays (CDAs) in which the interactions between factors are considered to be ordered. CDAs can be used to generate test suites for locating and detecting interaction faults between neighboring factors. We establish a general criterion for measuring the optimality of CDAs in terms of their size. Based on this optimality criterion, the equivalence between optimum CDAs and consecutive orthogonal arrays with prescribed properties is explored. Using the advantages of this equivalence, a great number of optimum CDAs are presented. In particular, the existence of optimum CDAs with few factors is completely determined. [ABSTRACT FROM AUTHOR]

We complete the classification of tight 4-designs in Hamming association schemes H(n,q), i.e., that of tight orthogonal arrays of strength 4, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight 4-design in H(n,q) and analyze its triple intersection numbers to conclude the non-existence in all open cases. [ABSTRACT FROM AUTHOR]

A (t, s, v)-all-or-nothing transform (AONT) is a bijective mapping defined on s-tuples over an alphabet of size v, which satisfies that if any s - t of the s outputs are given, then the values of any t inputs are completely undetermined. When t and v are fixed, to determine the maximum integer s such that a (t, s, v)-AONT exists is the main research objective. In this paper, we solve three open problems proposed in Nasr Esfahani et al. (IEEE Trans Inf Theory 64:3136–3143, 2018) and show that there do exist linear (2, p, p)-AONTs. Then for the size of the alphabet being a prime power, we give the first infinite class of linear AONTs which is better than the linear AONTs defined by Cauchy matrices. Besides, we also present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays. [ABSTRACT FROM AUTHOR]

We consider designs for cancer trials which allow each medical centre to treat only a limited number of cancer types with only a limited number of drugs. We specify desirable properties of these designs, and prove some consequences. Then we give several different constructions. Finally we generalize this to three or more factors, such as biomarkers. [ABSTRACT FROM AUTHOR]

A covering array of strength t on v symbols is an array with the property that, for every t-set of column vectors, every one of the $$v^t$$ possible t-tuples of symbols appears as a row at least once in the sub-array defined by these column vectors. Arrays constructed using m-sequences over a finite field possess many combinatorial properties and have been used to construct various combinatorial objects; see the recent survey Moura et al. (Des Codes Cryptogr 78(1):197-219, 2016). In this paper we construct covering arrays whose elements are the remainder of the division by some integer of the discrete logarithm applied to selected m-sequence elements. Inspired by the work of Colbourn (Des Codes Cryptogr 55(2-3):201-219, 2010), we prove our results by connecting the covering array property to a character sum, and we evaluate this sum by taking advantage of the balanced way in which the m-sequence elements are distributed. Our results include new infinite families of covering arrays of arbitrary strength. [ABSTRACT FROM AUTHOR]

A partial $$(k-1)$$ - spread in $${\text {PG}}(n-1,q)$$ is a collection of $$(k-1)$$ -dimensional subspaces with trivial intersection. So far, the maximum size of a partial $$(k-1)$$ -spread in $${\text {PG}}(n-1,q)$$ was known for the cases $$n\equiv 0\pmod k$$ , $$n\equiv 1\pmod k$$ , and $$n\equiv 2\pmod k$$ with the additional requirements $$q=2$$ and $$k=3$$ . We completely resolve the case $$n\equiv 2\pmod k$$ for the binary case $$q=2$$ . [ABSTRACT FROM AUTHOR]

This paper presents a general method of constructing schematic saturated orthogonal arrays of strength two. We consider a class of mixed saturated orthogonal arrays produced from saturated symmetric orthogonal arrays by using the contractive replacement method, study the Hamming distances of the rows, construct their association schemes, and prove that they are schematic. Some examples are given to illustrate this construction method. [ABSTRACT FROM AUTHOR]

A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. Since the paper of Hammons et al., there is a lot of interest in codes over rings, especially in codes over $$\mathbb {Z}_{4}$$ and their (usually non-linear) binary Gray map images. We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a $$\mathbb {Z}_{4}$$ code is one less than the minimum Lee weight of its Gray map image. [ABSTRACT FROM AUTHOR]

A mixed covering array (MCA) of type ( 1, v ,..., v ), denoted by MCA ( N; t, k, ( v , v ,..., v )), is an N × k array with entries in the i-th column from a set of v symbols and has the property that each N × t sub-array covers all the t-tuples at least λ times, where 1 ≤ i ≤ k. An MCA( N; t, k, ( 1, v ,..., v )) is said to be super-simple, if each of its N × ( t + 1) sub-arrays contains each ( t + 1)-tuple at most once. Recently, it was proved by Tang, Yin and the author that an optimum super-simple MCA of type ( a, b, b,..., b) is equivalent to a mixed detecting array (DTA) of type ( a, b, b,..., b) with optimum size. Such DTAs can be used to generate test suites to identify and determine the interaction faults between the factors in a component-based system. In this paper, some combinatorial constructions of optimum super-simple MCAs of type ( a, b, b,..., b) are provided. By employing these constructions, some optimum super-simple MCAs are then obtained. In particular, the spectrum across which optimum super-simple MCA(2 b ; 2, 4, ( a, b, b, b))′s exist, is completely determined, where 2 ≤ a ≤ b. [ABSTRACT FROM AUTHOR]

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $$t, k, \lambda , s$$ and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t- $$\mathrm {GHD}_{k}(s,v;\lambda )$$ , is an $$s\times s$$ array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than $$\lambda $$ cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that $$t=2$$ , $$k=3$$ and $$\lambda =1$$ , and write $$\mathrm {GHD}(s,v)$$ . In this case, the number of empty cells in each row and column falls between 0 and $$(s-1)/3$$ . Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least $$(s-2)/3$$ empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a $$\mathrm {GHD}(n+1,3n)$$ if and only if $$n \ge 6$$ , except possibly for $$n=6$$ . In the case of two empty cells, we show that there exists a $$\mathrm {GHD}(n+2,3n)$$ if and only if $$n \ge 6$$ . Noting that the proportion of cells in a given row or column of a $$\mathrm {GHD}(s,v)$$ which are empty falls in the interval [0, 1 / 3), we prove that for any $$\pi \in [0,5/18]$$ , there is a $$\mathrm {GHD}(s,v)$$ whose proportion of empty cells in a row or column is arbitrarily close to $$\pi $$ . [ABSTRACT FROM AUTHOR]

We present a novel secret sharing scheme where the secret is an autotopism (a symmetry) of a Latin square. Previously proposed secret sharing schemes involving Latin squares have many drawbacks: (a) Latin squares contain $$n^2$$ entries, which may be too large, (b) partial information about the secret may be directly revealed, (c) a subsequently discovered subtle 'flaw', (d) difficulty in initialization and reconstruction, (e) difficulty in verification, and (f) difficulty in generalizing to a multi-level scheme. We carefully analyze the security of the proposed scheme, and identify how it overcomes all of these problems. [ABSTRACT FROM AUTHOR]

Fuji-Hara and Kamimura (Util Math 43:65-70, 1993) outlined a method for constructing orthogonal arrays of strength 2 on the complement of a Baer subplane, with $$q(q-1)$$ symbols for a prime power $$q$$ . In this paper, we demonstrate that these orthogonal arrays can be decomposed into other orthogonal arrays of strength 2, with the same numbers of constraints and symbols but with smaller sizes and indices. In our construction, each orthogonal array of the decomposition can be obtained as an orbit of the point-set of a Baer subplane, under the action of a certain projective linear group. Furthermore, for $$q \equiv 2 \pmod 3$$ and $$q > 2$$ , a series of the new orthogonal arrays cannot be obtained by Bush's direct product construction, which is a classical method for constructing orthogonal arrays with non-prime-power numbers of symbols. [ABSTRACT FROM AUTHOR]

An equitable $$(r,c;v)$$ -rectangle is an $$r \times c$$ matrix $$L=(l_{ij})$$ with symbols from $$\mathbb {Z}_v$$ in which each symbol appears in every row either $$\left\lceil c/v \right\rceil $$ or $$\left\lfloor c/v \right\rfloor $$ times and in every column either $$\left\lceil r/v \right\rceil $$ or $$\left\lfloor r/v \right\rfloor $$ times. We call $$L$$ diagonally cyclic if $$l_{(i+1) (j+1)}=l_{ij}+1$$ , where the rows are indexed by $$\mathbb {Z}_r$$ and columns indexed by $$\mathbb {Z}_c$$ . We give a constructive proof of necessary and sufficient conditions for the existence of a diagonally cyclic equitable $$(r,c;v)$$ -rectangle. [ABSTRACT FROM AUTHOR]

By using the relationship between orthogonal arrays and decompositions of projection matrices and projection matrix inequalities, we present a method for constructing a class of new orthogonal arrays which have higher percent saturations. As an application of the method, many new mixed-level orthogonal arrays of run sizes 108 and 144 are constructed. [ABSTRACT FROM AUTHOR]

This paper considers the use of orthogonal arrays of strength two as experimental designs for fitting a surrogate model. Contrary to standard space-filling designs or Latin hypercube designs, the points of an orthogonal array of strength two are well distributed when they are projected on the two-dimensional faces of the unit cube. The aim is to determine if this property allows one to fit an accurate surrogate model when the computer response is governed by second-order interactions of some input variables. The first part of the paper is devoted to the construction of orthogonal arrays with space-filling properties. In the second part, orthogonal arrays are compared with standard designs for fitting a Gaussian process model. [ABSTRACT FROM PUBLISHER]

Nowadays orthogonal arrays play important roles in statistics and other fields. Usual difference matrices are essential for the construction of many symmetrical or a few asymmetrical orthogonal arrays. But there are also asymmetrical orthogonal arrays which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices were discovered from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented. As an application, a lot of new orthogonal arrays of run size 100 have been constructed. [ABSTRACT FROM AUTHOR]

It was proved recently that a super-simple orthogonal array (SSOA) of strength $$t$$ and index $$\lambda \ge 2$$ is equivalent to a minimum detecting array (DTA). In computer software tests in component-based systems, such a DTA can be used to generate test suites that are capable of locating $$d=\lambda -1$$ $$t$$-way interaction faults and detect whether there are more than $$d$$ interaction faults. It is proved in this paper that an SSOA of strength $$t=3$$, index $$\lambda \ge 2$$ and degree $$k=5$$, or an SSOA $$_{\lambda }(3,5,v)$$, exists if and only if $$\lambda \le v$$ excepting possibly a handful of cases. [ABSTRACT FROM AUTHOR]

Minimum aberration designs are preferred in practice, especially when it is desired to carry out a multi-factor experiment using less number of runs. Several authors considered constructions of minimum aberration designs. Some used computer algorithms and some listed good designs from the exhausted search. We propose a simple method to obtain minimum aberration designs for experiments of size less than or equal to thirty-two. Here, we use an ordered sequence of columns from an orthogonal array to design experiments and blocked experiments. When the method is implemented in MS Excel, minimum aberration designs can be easily achieved. [ABSTRACT FROM AUTHOR]

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA (27,13,3,2), an OA (18,7,3,2) and an OA (36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D -efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model. [ABSTRACT FROM AUTHOR]