Coding‐theoretic constructions for (t,m,s)‐nets and ordered orthogonal arrays

(t,m,s)‐nets are point sets in Euclidean s‐space satisfying certain uniformity conditions, for use in numerical integration. They can be equivalently described in terms of ordered orthogonal arrays, a class of finite geometrical structures generalizing orthogonal arrays. This establishes a link between quasi‐Monte Carlo methods and coding theory. The ambient space is a metric space generalizing the Hamming space of coding theory. We denote it by NRT space (named after Niederreiter, Rosenbloom and Tsfasman). Our main results are generalizations of coding‐theoretic constructions from Hamming space to NRT space. These comprise a version of the Gilbert‐Varshamov bound, the (u,u+υ)‐construction and concatenation. We present a table of the best known parameters of q‐ary (t,m,s)‐nets for qε{2,3,4,5} and dimension m≤50. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 403–418, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10015     

Constructions of nets via OAs of strength 3 with a prescribed property

The idea of (t, m, s)‐nets was proposed by Niederreiter in 1987. Such nets are highly uniform point distributions in s‐dimensional unit cubes and useful in numerical analysis. It is by now well known that (t, m, s)‐nets can be equivalently described in terms of ordered orthogonal arrays (OOAs). In this article, we describe an equivalence between an OOA and an orthogonal array (OA) with all its derived orthogonal subarrays being resolvable. We then present a number of constructions for OAs where all their derived orthogonal subarrays are resolvable. These results are finally combined to give new series of (t, m, s)‐nets. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:144‐155, 2011     

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.     

A combinatorial characterization of (t,m,s)-nets in base b

We consider (t,m,s)-nets in base b, which were introduced by Niederreiter in 1987. These nets are highly uniform point distributions in s-dimensional unit cubes and have applications in the theory of numerical integration and pseudorandom number generation. A central question in their study is the determination of the parameter values for which these nets exist. Niederreiter has given several methods for their construction, all of which are based on a general construction principle from his 1987 paper. We define a new family of combinatorial objects, the so-called “generalized orthogonal arrays,” and then discuss a combinatorial characterization of (t.m.s)-nets in base b in terms of these generalized orthogonal arrays. Using this characterization, we describe a new method for constructing (t.m.s)-nets in base b that is not based on the aforementioned construction principle. This method gives rise to some very general conditions on the parameters (involving a link with the theory of orthogonal arrays) that are sufficient to ensure the existence of a (t.m.s)-net in base b. In this way we construct many nets that are new. © 1996 John Wiley & Sons, Inc.     

The triple distribution of codes and ordered codes

We study the distribution of triples of codewords of codes and ordered codes. Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (8) (2005) 2859–2866] used the triple distribution of a code to establish a bound on the number of codewords based on semidefinite programming. In the first part of this work, we generalize this approach for ordered codes. In the second part, we consider linear codes and linear ordered codes and present a MacWilliams-type identity for the triple distribution of their dual code. Based on the non-negativity of this linear transform, we establish a linear programming bound and conclude with a table of parameters for which this bound yields better results than the standard linear programming bound.     

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     

Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.     

Construction of digital nets based on propagation rules for OOAs

(t,m,s)-nets are a powerful tool for the generation of low-discrepancy point sets. We find nets with improved parameters using a coding-theoretic approach, starting with an orthogonal array (dual of a linear code) and embedding it in a net. The innovation of our approach is that propagation rules for OOAs are applied halfway through the embedding process. To this end many propagation rules have been generalized from the setting of orthogonal arrays to OOAs. According to MinT [1], the online database of (t,m,s)-net parameters at mint.sbg.ac.at , more than 100 nets with previously unknown parameters have been found using this method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The decomposability of simple orthogonal arrays on 3 symbols having t + 1 rows and strength t

It is well‐known that all orthogonal arrays of the form OA(N, t + 1, 2, t) are decomposable into λ orthogonal arrays of strength t and index 1. While the same is not generally true when s = 3, we will show that all simple orthogonal arrays of the form OA(N, t + 1, 3, t) are also decomposable into orthogonal arrays of strength t and index 1. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 442–458, 2000     

Linear hash families and forbidden configurations

The classical orthogonal arrays over the finite field underlie a powerful construction of perfect hash families. By forbidding certain sets of configurations from arising in these orthogonal arrays, this construction yields previously unknown perfect, separating, and distributing hash families. When the strength s of the orthogonal array, the strength t of the hash family, and the number of its rows are all specified, the forbidden sets of configurations can be determined explicitly. Each forbidden set leads to a set of equations that must simultaneously hold. Hence computational techniques can be used to determine sufficient conditions for a perfect, separating, and distributing hash family to exist. In this paper the forbidden configurations, resulting equations, and existence results are determined when (s, t) ∈ {(2, 5), (2, 6), (3, 4), (4, 3)}. Applications to the existence of covering arrays of strength at most six are presented.      

Cubic and higher degree bounds for codes and (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-nets

The Plotkin bound and the quadratic bound for codes and (t, m, s)-nets can be obtained from the linear programming bound using certain linear and quadratic polynomials, respectively. We extend this approach by considering cubic and higher degree polynomials to find new explicit bounds as well as new non-existence results for codes and (t, m, s)-nets.     

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

On the concentration of multivariate polynomials with small expectation

Let t₁,…,t_n be independent, but not necessarily identical, {0, 1} random variables. We prove a general large deviation bound for multivariate polynomials (in t₁,…,t_n) with small expectation [order O(polylog(n))]. Few applications in random graphs and combinatorial number theory will be discussed. Our result is closely related to a classical result of Janson [Random Struct Algorithms 1 (1990), 221–230]. Both of them can be applied in similar situations. On the other hand, our result is symmetric, while Janson's inequality only deals with the lower tail probability. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 344–363, 2000     

A Note on Geometric Structures of Linear Ordered Orthogonal Arrays and (T,M,S)-Nets of Low Strength

The equivalent geometrical configurations of linear ordered orthogonal arrays are determined when their strengths are 3 and 4. Existence of such geometrical configurations is investigated. They are also useful in the study of (T, M, S)-nets.     

New Bounds for Perfect Hashing via Information Theory

A set of sequences of length t from a b-element alphabet is called k-separated if for every k-tuple of the sequences there exists a coordinate in which they all differ. The problem of finding, for fixed t, b, and k, the largest size N(t, b, k) of a k-separated set of sequences is equivalent to finding the minimum size of a (b, k)-family of perfect hash functions for a set of a given size. We shall improve the bounds for N(t, b, k) obtained by Fredman and Komlós [1].Körner [2] has shown that the proof in [1] can be reduced to an application of the sub-additivity of graph entropy [3]. He also pointed out that this sub-additivity yields a method to prove non-existence bounds for graph covering problems. Our new non-existence bound is based on an extension of graph entropy to hypergraphs.     

Semidefinite approximations for quadratic programs over orthogonal matrices

Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from \mathbbR^n×k{\mathbb{R}^{n\times k}} , then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Graph partitioning problem (GPP), the Quadratic assignment problem (QAP) etc. In particular we show how to improve significantly the well-known Donath-Hoffman eigenvalue lower bound for GPP by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for GPP and QAP yields the exact values.     

Covering arrays of higher strength from permutation vectors

A covering array CA(N;t,k,v) is an N × k array such that every N × t sub‐array contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. We introduce a combinatorial technique for their construction, focussing on covering arrays of strength 3 and 4. With a computer search, covering arrays with improved parameters have been found. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 202–213, 2006     

Preemptive scheduling and antichain polyhedra

We present a theoretical framework, which is based upon notions of ordered hypergraphs and antichain polyhedra, and which is dedicated to the combinatorial analysis of preemptive scheduling problems submitted to parallelization constraints.This framework allows us to characterize specific partially ordered structures which are such that induced preemptive scheduling problems may be solved through linear programming. To prove that, in the general case, optimal preemptive schedules may be searched inside some connected subset of the vertex set of an Antichain Polyhedron.     

Contribution of copositive formulations to the graph partitioning problem

This article provides analysis of several copositive formulations of the graph partitioning problem and semidefinite relaxations based on them. We prove that the copositive formulations based on results from Burer [S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–495] and the author of the paper [J. Povh, Semidefinite approximations for quadratic programs over orthogonal matrices. J. Global Optim. 48 (2010), pp. 447–463] are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound [W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 (1973), pp. 420–425] and the projected semidefinite lower bound from Wolkowicz and Zhao [H. Wolkowicz and Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97 (1999), pp. 461–479].