Rebeka Kiss, Gábor P. Nagy
We show that no orthogonal arrays $OA(16 λ, 11, 2,4)$ exist with $λ=6$ and $λ=7$. This solves an open problem of the NSUCRYPTO Olympiad 2018. Our result allows us to determine the minimum weights of certain higher-order correlation-immune Boolean functions.
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We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $k\leq 11$, and we present a general existence result that holds for all $k \geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.
A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array $\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\}$ ($r, t \ge 1$), $\{135, 128, 16; 1, 16, 120\}$, $\{234, 165, 12; 1, 30, 198\}$ or $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are…
New results on large sets of orthogonal arrays and orthogonal arrays
该研究针对组合设计领域的正交阵列及其大集问题，通过多种构造方法得到新的正交阵列大集系列，再结合Kronecker积获得大量新的无穷类正交阵列，拓展了相关研究成果。
Riordan arrays, orthogonal polynomials as moments, and Hankel transforms
以Legendre和Hermite正交多项式为例，说明了如何解释这些正交多项式是其它正交多项式的矩，并给出了它们的Riordan阵列.我们使用这些手段来计算相关的多项式序列的汉克尔变换。
For prime powers q we use "strongly orthogonal" linear Sudoku solutions of order q^2 to construct ordered orthogonal arrays of type OOA (4,s,2,q), and for each q we present a range of values of s for which these constructions are valid.
We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.
A linear algebraic approach to orthogonal arrays and Latin squares
为了研究正交表和符号正交表，Ray-Chaudhuri和Singhi（1988和1994）考虑了一些模空间。在这里，使用线性代数方法，我们定义了一个包含矩阵，并找到它的秩。在特殊的情况下，拉丁方，我们表明，有一个简单的算法生成一个基础，这个矩阵使用所谓的intercalates。我们还扩展了最后一个想法。
Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.
In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In particular we provide explicit constructions of such decompositions for cycles of length $k\leq 10$. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.
In this expository paper, we mainly study orthogonal arrays (OAs) of strength two having a row that is repeated $m$ times. It turns out that the Plackett-Burman bound (\cite{PB}) can be strengthened by a factor of $m$ for orthogonal arrays of strength two that contain a row that is repeated $m$ times. This is a consequence of a more general result due to Mukerjee, Qian and Wu \cite{Muk} that applies to orthogonal arrays of arbitrary strength $t$. We examine several proofs of the Plackett-Burman bound and discuss which of these proofs can be strengthened to yield the aforementioned bound for OA…
We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.
Large set of orthogonal arrays (LOA) were introduced by D. R. Stinson, and it is also used to construct multimagic squares recently. In this paper, multimagic squares based on strong double LOA are further investigated. It is proved that there exists an MS$(q^{2t-1},t)$ for any prime power $q\geq 2t-1$ with $t\geq3$, which provided a new family of multimagic squares.
In this paper, we show how certain three-class association schemes and orthogonal arrays give rise to partial geometric designs. We also investigate the connections between partial geometric designs and certain regular graphs having three or four distinct eigenvalues, three-class association schemes, orthogonal arrays of strength two and particular linear codes. We give various characterizations of these graphs, association schemes and orthogonal arrays in terms of partial geometric designs. We also give a list of infinite families of directed strongly regular graphs arising from the partial g…
Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.
Counting patterns in colored orthogonal arrays
设$S$是一个正交数组$OA(d,k)$，$c$是它的基集$X$的$r$ -染色.我们给出了一个组合恒等式，它将$S$中的向量数与$c$下给定的颜色模式与颜色类的基数联系起来。几个应用程序的身份被认为是。其中，我们证明了整数区间$[1,n]$的每一个均匀$r$ --染色至少有$1/2(n/r)^2+O(n)$单色Schur三元组。我们还表明，在一个正交表$OA(d,d-1)$，单色矢量的每种颜色的数量只取决于错过该颜色的矢量的数量和颜色类的基数。
It is known that correlation-immune (CI) Boolean functions used in the framework of side-channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao's Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple…
In \cite{Castoldi}, $q^t \by (q+1)t$ ordered orthogonal arrays (OOAs) of strength $t$ over the alphabet $\FF_q$ were constructed using linear feedback shift register sequences (LFSRs) defined by {\em primitive} polynomials in $\FF_q[x]$. In this paper we extend this result to all polynomials in $\FF_q[x]$ which satisfy some fairly simple restrictions, restrictions that are automatically satisfied by primitive polynomials. While these restrictions sometimes reduce the number of columns produced from $(q+1)t$ to a smaller multiple of $t$, in many cases we still obtain the maximum number of colum…
Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and augmented orthogonal arrays (Discrete Math. 341 (2018), 299-307). In this paper, we show that there is an AOA$(s,t,k,v)$ if and only if there is an OA$(t,k,v)$ which can be partitioned into $v^{t-s}$ subarrays, each being an OA$(s,k,v)$, and that there is a linear AOA$(s,t,k,q)$ if and only if there is a linear maximum distance separable (MDS) code of length $k$ and dimension $t$ over $\mathbb{F}_q$ which contains a linear MDS subcode of length $k$ and dimension $s$ over $\m…
Symmetry plays a fundamental role in design of experiments. In particular, symmetries of factorial designs that preserve their statistical properties are exploited to find designs with the best statistical properties. By using a result proved by Rosenberg [6], the concept of the LP relaxation orthogonal array polytope is developed and studied. A complete characterization of the permutation symmetry group of this polytope is made. Also, this characterization is verified computationally for many cases. Finally, a proof is provided.
Extremal orthogonal arrays
本文研究极值正交数组的性质，证明其诱导分裂方案，并获得紧正交数组存在性的新条件。
$d$-orthogonal polynomials, Fuss-Catalan matrices and lattice paths
本文利用$d$-正交多项式定义Fuss-Catalan-Riordan阵列，并建立其与Fuss Catalan数和格路的联系，强调生产矩阵的作用。
On Construction of a Class of Orthogonal Arrays (Thesis)
提出了一种构造正交表的新方法。该算法利用Kronecker乘积算子和单位列向量从已有的正交表中生成新的正交表。该算法的有效性在于，它适用于任何线性种子正交表，而不施加任何限制的强度或水平的数量。得到的正交表具有与种子正交表相同的强度。我们还讨论了算法正确性的证明。在结果部分中，我们提供了一个使用该算法生成的新正交表列表，这些正交表当前没有出现在正交表库中。
We construct orthogonal arrays OA$_λ (k,n)$ (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 $m / λ$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n+1$, albeit with large $λ$. We also study basic OAs; these are optimal OAs in which $\gcd(m,λ) = 1$. 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 wth $n=2$, modulo the Hadam…
Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences
研究具多项式系数线性齐次递推的多项式序列的Riordan阵列表示，刻画二阶递推情形并推广到任意阶；构建任意正交多项式序列的生成矩阵框架，还将其拓展至$d$-正交多项式序列，揭示对应递推与矩序列等性质。
We study properties of constant coefficient Laurent biorthogonal polynomials using Riordan arrays. We give details of related orthogonal polynomials, and we explore relationships between the moments of these orthogonal polynomials, the moments of the defining Laurent biorthogonal polynomials, and the expansions of $T$-fractions. Closed form expressions are given for the polynomials and their moments.
Using the theory of exponential Riordan arrays, we show that the Eulerian-Dowling polynomials are moments for a paramaterized family of orthogonal polynomials. In addition, we show that the related Dowling and the Tanny-Dowling polynomials are also moments for appropriate families of orthogonal polynomials. We provide continued fraction generating functions and Hankel transforms for these polynomials.
We characterize mixed-level orthogonal arrays in terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays,…
We derive explicit formulas for the resultants and discriminants of classical quasi-orthogonal polynomials, as a full generalization of the results of Dilcher and Stolarsky (2005) and Gishe and Ismail (2008). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermi…
Mixed orthogonal arrays, $k$-dimensional $M$-part Sperner multi-families, and full multi-transversals
Aydinian等人[J. Combinatorial Theory A 118（2）（2011），702-725]用$(m_1,m_2,...,m_M;L_1,L_2,...,L_M)$型M部分Sperner族的一组M不等式代替了L-Sperner族的通常BLYM不等式，并证明了如果所有不等式都相等，则该族是齐次的。Aydinian等人[Australasian J. Comb. 48（2010），133-141]观察到所有不等式都成立，当且仅当Sperner族的截线对应于具有约束M，强度M-1的简单混合正交表，在$i^{\text{th}}$列中使用$m_i+1$符号。本文定义了含参数$L_P: P\in\binom{[M]}{k}$的$k$ -维$M$ -部分Sperner多族，并证明了它们的$\binom{M}{k}$ BLYM不等式.证明了当k<M且所有不等式成立且相等时，这类多族必是齐次的，其剖面矩阵是强度为M-k的混合正交表.对于k=M，齐性不一定成立，但对某些单族给出了一些必要条件。遵循Aydinian等人的方法[Australasian J. Comb.48（2010），133-141]，我们在第i列中使用$m_i+1$符号，给出了具有约束M，强度M-k的简单混合正交表的新构造。我们扩展的凸壳方法k维M部分Sperner多家庭，并允许额外的条件提供新的结果，即使是…
Eulerian polynomials as moments, via exponential Riordan arrays
利用指数Riordan阵列和正交多项式的理论，证明了“降幂“欧拉多项式及其一次移位序列是简单正交多项式族的矩序列，并利用它们的三项递归性刻画了它们的矩序列.得到了多项式序列的连分数生成函数，并计算了它们的Hankel变换。
Using the theory of exponential Riordan arrays, we show that the $1/k$-Eulerian polynomials are moments for a paramaterized family of orthogonal polynomials. In addition, we show that the related Savage-Viswanathan polynomials are also moments for appropriate families of orthogonal polynomials. We provide continued fraction ordinary generating functions and Hankel transforms for these moments, as well as the three-term recurrences for the corresponding orthogonal polynomials. We provide formulas for the $1/k$-Eulerian polynomials and the Savage-Viswanathan polynomials involving the Stirling nu…
Almost Orthogonal Arrays: Search Three Ways
本文通过整数规划、局部搜索元启发式和代数方法三种方式寻找近正交数组，并展示了其在非正交性度量上的改进。
Improved Bounds for Nested Orthogonal Arrays
本文在群论框架下重新表述Rao界，推导非对称嵌套正交阵列（NOAs）的更紧下界，其在因子水平相等时归约为对称界，且通过构造NOAs验证了新 bounds 的最优性。
The focus of this thesis is the study and construction of covering arrays, relying on maximal period sequences and other tools from finite fields. A covering array of strength $t$, denoted $\mathrm{CA}(N; t, k,v)$, is an $N\times k$ array with entries from an alphabet $A$ of size $v$, with the property that in the $N\times t$ subarray defined by any $t$ columns, each of the $v^t$ vectors in $A^t$ appears at least once as a row. Covering arrays generalize orthogonal arrays, which are classic combinatorial objects that have been studied extensively. Constructing covering arrays with a small row-…
Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods based on finding all non-isomorphic solutions of two novel integer linear programming formulations for classifying all non-isomorphic OA(N,k,s,t) given a set of all non-isomorphic OA(N,k-1,s,t). We also establish the concept of orthogonal design equivalence of OA(N,k,2,t) to reduce the number of integer linear programs (ILPs) whose all non-isomorphic soluti…
With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)…
Orthogonal arrays from Hermitian varieties
利用PGL（n+1，q^2）的一个子集对PG（n，q^2）中某些非退化Hermite簇的作用，构造了一个正交表OA（q^{2n-1}，q^{2n-2}，q，2）.它还表明，该正交表的行对应于仿射设计的一些块，这对于q> 2是一个非经典模型的仿射空间AG（2n-1，q）。
We study the classes of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences, which are generalizations of $(u,m,s)$-nets and $(u,s)$-sequences, respectively. We show equivalence results that link the existence of $(u,m,{\bf e},s)$-nets and so-called mixed (ordered) orthogonal arrays, thereby generalizing earlier results by Lawrence, and Mullen and Schmid. We use this combinatorial equivalence principle to obtain new results on the possible parameter configurations of $(u,m,{\bf e},s)$-nets and $(u,{\bf e},s)$-sequences, which generalize in particular a result of Martin and Stinson.
Bounds on ordered codes and orthogonal arrays
我们得到新的估计的大小的代码和正交表的有序汉明空间（Niederreiter-Rosenbloom-Tsfasman空间）。我们还表明，有序的汉明计划，结合计划，描述组合的空间，给出的特征值的多变量Krawtchouk多项式，并建立他们的一些性质。
On quasi-Hermitian varieties in even characteristic and related orthogonal arrays
本文研究特征2、维数3的BM准埃尔米特簇，证明其射影等价性并完成分类，确定其直射变换群结构，还构造了一族元素取自GF(q)的简单正交阵列O(q⁵,q⁴,q,2)。
Parallelizing the branch-and-bound with isomorphism pruning algorithm for classifying orthogonal arrays
本文提出并行化分支限界与同构剪枝算法，用于分类正交数组，实现线性加速并首次完成特定参数的分类任务。
Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of $N$ subsystems with an arbitrary number of internal levels each, based on properties of orthogonal arrays with $N$ columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four…
Local unitary equivalence of orthogonal arrays and related linear codes
本文通过生成矩阵构造正交阵列，建立了基于傅里叶的局部酉等价条件，并揭示了其与线性码（尤其是偶校验码）之间的显式联系，识别了由特定线性码产生的等价族。
Hilbert Bases for Orthogonal Arrays
本文将生成给定维数和力的所有二级正交表，即OA $(n,m)$中的元素的问题与一个包含有理凸锥的整数规划问题的求解联系起来，其中$n$是因子数，$m$是力。我们不限制数组中的点数，也就是说，我们允许任何数量的复制。这个问题在理论上可以通过希尔伯特基来解决，希尔伯特基形成无限集OA $(n,m)$中所有元素的有限生成集。我们讨论了一些例子，明确解决了一个软件进行希尔伯特基计算。
Generalizations of nets and Latin squares
研究推广(k,n)-网等的组合结构，通过网状结构定义协作系统与半正交阵列，引入副同构和同构概念，给出小网状结构构造及大网状结构的三种递归构造。
Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters
该研究提出首个适用于所有参数的显式确定性近最优规模正交阵列构造方法，基于代数几何码，还可生成任意定义域与陪域的高效t独立哈希函数，解决了此前仅存非构造性证明的问题。
The modular design of planar phased arrays arranged on orthogonal polygon-shaped apertures is addressed and a new method is proposed to synthesize domino-tiled arrays fitting multiple, generally conflicting, requirements. Starting from an analytic procedure to check the domino-tileability of the aperture, two multi-objective optimization techniques are derived to efficiently and effectively deal with small and medium/large arrays depending on the values of the bounds for the cardinality of the solution space of the admissible clustered solutions. A set of representative numerical examples is r…
The classification of orthogonal arrays OA(2048,14,2,7) and some completely regular codes
本文对 OA(2048,14,2,7) 及其等价的 14 维立方体完全正则码进行完全分类，并确定派生正交表和相关完全正则码的等价类数量。
We characterize certain Riordan arrays by their $A$-matrices and $ρ$ sequences. We conjecture the form of a generic $A$-matrix which leads to Somos $4$ sequences. We find an $A$-matrix that produces a Riordan quasi-involution, and we study the $A$-matrices and $ρ$ sequences of the moment matrices of certain perturbed orthogonal polynomials.
The Lattice of N-Run Orthogonal Arrays
如果指定了强度为2的（混合水平）正交数组中的运行游程数，则可能有多少水平和因子？具有N次运行的正交表的可能参数集的集合具有自然的晶格结构，这是由“扩展的构造”方法引起的。特别是在这个晶格中的对偶原子是最重要的参数集，因为任何其他参数集的N运行正交阵列可以构造从他们。为了了解对偶原子的数量，并开始理解晶格作为N的函数，我们研究晶格的高度和大小。结果表明，该系统的高度至多为[c（N-1）]，其中c= 1.4039.并且存在N的值的无穷序列，对于该序列，达到该界。另一方面，格中的节点数由N的超多项式函数限定（对于N的某些值序列，超多项式增长确实会发生）。使用基于"混合分布“的新构造，确定了具有64次运行的所有参数集。其中四个64运行正交表似乎是新的。
It is an elementary fact that the size of an orthogonal array of strength t on k factors must be a multiple of a certain number, say L_t, that depends on the orders of the factors. Thus L_t is a lower bound on the size of arrays of strength t on those factors, and is no larger than L_k, the size of the complete factorial design. We investigate the relationship between the numbers L_t, and two questions in particular: For what t is L_t < L_k? And when L_t = L_k, is the complete factorial design the only array of that size and strength t? Arrays are assumed to be mixed-level. We refer to an arra…
There is always a natural embedding of $S_s\wr S_k$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$ level, strength $1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_2\wr S_k$ for all $k$, and in the $2$ level, strength $2$ case it is isomorphic to $S_2^k\rtimes S_{k+1}$ for $k\geq 4$. The strength $2$ result reveals previously unknown permutation symmetries that can not be captured …
In 1996, Jackson and Martin proved that a strong ideal ramp scheme is equivalent to an orthogonal array. However, there was no good characterization of ideal ramp schemes that are not strong. Here we show the equivalence of ideal ramp schemes to a new variant of orthogonal arrays that we term augmented orthogonal arrays. We give some constructions for these new kinds of arrays, and, as a consequence, we also provide parameter situations where ideal ramp schemes exist but strong ideal ramp schemes do not exist.
A pair of orthogonal latin cubes of order $q$ is equivalent to an MDS code with distance $3$ or to an ${\rm OA}_1(3,5,q)$ orthogonal array. We construct pairs of orthogonal latin cubes for a sequence of previously unknown orders $q_i=16(18i-1)+4$ and $q'_i=16(18i+5)+4$. The minimal new obtained parameters of orthogonal arrays are ${\rm OA}_1(3,5,84)$. Keywords: latin square, latin cube, MOLS, MDS code, block design, Steiner system, orthogonal array
We present a new construction of ordered orthogonal arrays (OOA) of strength $t$ with $(q + 1)t$ columns over a finite field $\mathbb{F}_{q}$ using linear feedback shift register sequences (LFSRs). OOAs are naturally related to $(t, m, s)$-nets, linear codes, and MDS codes. Our construction selects suitable columns from the array formed by all subintervals of length $\frac{q^{t}-1}{q-1}$ of an LFSR sequence generated by a primitive polynomial of degree $t$ over $\mathbb{F}_{q}$. We prove properties about the relative positions of runs in an LFSR which guarantee that the constructed OOA has str…
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. We show that Delsarte's observation extends to codes over arbitrary finite rings. Since the paper of Hammons \emph{et al.}, there is a lot of interest in codes over rings, especially in codes over $Z_4$ 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 $Z_4$ co…
We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating functions representable as continued fractions. Links to orthogonal polynomial moment sequences, and to Laurent biorthogonal polynomials are developed. We show that certain Riordan group involutions can be defined by continued fractions. We also show how simple transformations of the Jacobi continued fractions can lead to exponential Riordan arrays. Finally, by way…
In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the so…
For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by $G^{\rm LP}$, is the symmetry group of the LP. Margot [F. Margot, 50 Years of Integer Programming 1958-2008 (2010), 647-686] described a method for computing a subgroup of the symmetry group $G^{\rm LP}$ of an LP. This method computes $G^{\rm LP}$ when the LP has only non-redundant inequalities and its feasible set satisfies no equality constr…
By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points polytope of an orthogonal array (OA) defining integer linear program (ILP). Our results address the conjecture that if this polytope is non-empty, then it is full dimensional within the affine space where all the feasible points of the ILP's linear programming (LP) relaxation lie, raised by Appa et al., [On multi-index assignment polytopes, Linear Algebra and its Applications 416 (2-3) (2006), 224--241]. In particular, our theoretical results provide a su…
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