An information-theoretic method in combinatorial theory

Two examples are given showing the utility of Shannon's concepts of entropy and mutual information in combinatorial theory.     

Incidence matrices and inequalities for combinatorial designs

In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002     

A census of highly symmetric combinatorial designs

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v,k,1) designs, mainly for t=2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t>2 most of these characterizations have remained long-standing challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 3≤t≤6 is of particular interest and has been open for about 40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of Incidence Geometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably dating back to 1965). The present paper continues the author’s work (see Huber (J. Comb. Theory Ser. A 94, 180–190, 2001; Adv. Geom. 5, 195–221, 2005; J. Algebr. Comb., 2007, to appear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.      

A generalization of combinatorial designs and related codes

In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the t-adesign, which was coined by Ding (Codes from difference sets, 2015). It is clear that 2-adesigns are partially balanced incomplete block designs which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of 2-adesigns (some of which correspond to new almost difference sets and some to new almost difference families), as well as two constructions of 3-adesigns. We discuss basic properties of the incidence matrices and make an initial investigation into the codes which they generate. We find that many of the codes have good parameters in the sense they are optimal or have relatively high minimum distance.     

Good equidistant codes constructed from certain combinatorial designs

An (n,M,d;q) code is called equidistant code if the Hamming distance between any two codewords is d. It was proved that for any equidistant (n,M,d;q) code, d?nM(q-1)/(M-1)q(=d_opt, say). A necessary condition for the existence of an optimal equidistant code is that d_opt be an integer. If d_opt is not an integer, i.e. the equidistant code is not optimal, then the code with d=⌊d_opt⌋ is called good equidistant code, which is obviously the best possible one among equidistant codes with parameters n,M and q. In this paper, some constructions of good equidistant codes from balanced arrays and nested BIB designs are described.     

A problem of combinatorial designs related to authentication codes

The strong partially balanced t-designs can be used to construct authentication codes, whose probabilities P_r of successful deception in an optimum spoofing attack of order r for r = 0, 1, …, t − 1, achieve their information-theoretic lower bounds. In this paper a new family of strong partially balanced t-designs are constructed by means of rational normal curves over finite fields. Thus based on this new partially balanced t-designs a new class of authentication codes is obtained. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 417–429, 1998     

A link between combinatorial designs and three-weight linear codes

In this paper, we show that partial geometric designs can be constructed from certain three-weight linear codes, almost bent functions and ternary weakly regular bent functions. In particular, we show that existence of a family of partial geometric difference sets is equivalent to existence of a certain family of three-weight linear codes. We also provide a link between ternary weakly regular bent functions, three-weight linear codes and partial geometric difference sets.     

An information-theoretic framework for robustness

This is a paper about the foundation of robust inference. As a specific example, we consider semiparametric location models that involve a shape parameter. We argue that robust methods result via the selection of a representative shape from a set of allowable shapes. To perform this selection, we need a measure of disparity between the true shape and the shape to be used in the inference. Given such a disparity, we propose to solve a certain minimax problem. The paper discusses in detail the use of the Kullback-Leibler divergence for the selection of shapes. The resulting estimators are shown to have redescending influence functions when the set of allowable shapes contains heavy-tailed members. The paper closes with a brief discussion of the next logical step, namely the representation of a set of shapes by a pair of selected shapes.     

Self-dual codes from orbit matrices and quotient matrices of combinatorial designs

In this paper we give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by the action of an automorphism group of the design. Further, we give some further constructions of self-dual codes from symmetric block designs and their orbit matrices. Moreover, in a similar way as for symmetric designs, we give constructions of self-dual codes from SGDDs with the dual property and their quotient matrices.     

Linear spaces and transversal designs: k-anonymous combinatorial configurations for anonymous database search notes

Anonymous database search protocols allow users to query a database anonymously. This can be achieved by letting the users form a peer-to-peer community and post queries on behalf of each other. In this article we discuss an application of combinatorial configurations (also known as regular and uniform partial linear spaces) to a protocol for anonymous database search, as defining the key-distribution within the user community that implements the protocol. The degree of anonymity that can be provided by the protocol is determined by properties of the neighborhoods and the closed neighborhoods of the points in the combinatorial configuration that is used. Combinatorial configurations with unique neighborhoods or unique closed neighborhoods are described and we show how to attack the protocol if such configurations are used. We apply k-anonymity arguments and present the combinatorial configurations with k-anonymous neighborhoods and with k-anonymous closed neighborhoods. The transversal designs and the linear spaces are presented as optimal configurations among the configurations with k-anonymous neighborhoods and k-anonymous closed neighborhoods, respectively.     

New constructions for perfect hash families and related structures using combinatorial designs and codes

In this paper, we consider explicit constructions of perfect hash families using combinatorial methods. We provide several direct constructions from combinatorial structures related to orthogonal arrays. We also simplify and generalize a recursive construction due to Atici, Magliversas, Stinson and Wei [3]. Using similar methods, we also obtain efficient constructions for separating hash families which result in improved existence results for structures such as separating systems, key distribution patterns, group testing algorithms, cover‐free families and secure frameproof codes. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:189–200, 2000     

Geometric Measures for Random Curved Mosaics of Rd

This paper deals with stationary random mosaics of R^d with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R² and R³ is given.     

An information-theoretic approach to effective inference for Z-functionals

In this paper, through an information-theoretic approach, we construct estimations and confidence intervals of Z-functionals involving finite population and with the presence of auxiliary information. In particular, we give a method of estimating the variance of finite population with known mean. The modified estimates and confidence intervals for Z-functionals can adequately use the auxiliary information, at least not worse than what the standard ones do. A simulation study is presented to assess the performance of the modified estimates for the finite sample case.     

Inequalities for combinatorial sums

For \(k,l\in \mathbf {N}\), let

$$\begin{aligned}&P_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k-1} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }\\&\quad \text{ and }\quad Q_{k,l}=\Bigl (\frac{l}{k+l}\Bigr )^{k+l} \sum _{\nu =0}^{k} {k+l\atopwithdelims ()\nu } \Bigl (\frac{k}{l}\Bigr )^{\nu }. \end{aligned}$$

We prove that the inequality

$$\begin{aligned} \frac{1}{4}\le P_{k,l} \end{aligned}$$

is valid for all natural numbers k and l. The sign of equality holds if and only if \(k=l=1\). This complements a result of Vietoris, who showed that

$$\begin{aligned} P_{k,l}<\frac{1}{2} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

An immediate corollary is that

$$\begin{aligned} \frac{1}{4}\le P_{k,l}<\frac{1}{2} <Q_{k,l}\le \frac{3}{4} \quad {(k,l\in \mathbf {N})}. \end{aligned}$$

The constant bounds are sharp.

     

Constructions for orthogonal designs using signed group orthogonal designs

Craigen introduced and studied signed group Hadamard matrices extensively and eventually provided an asymptotic existence result for Hadamard matrices. Following his lead, Ghaderpour introduced signed group orthogonal designs and showed an asymptotic existence result for orthogonal designs and consequently Hadamard matrices. In this paper, we construct some interesting families of orthogonal designs using signed group orthogonal designs to show the capability of signed group orthogonal designs in generation of different types of orthogonal designs.     

Exact methods for combinatorial auctions

This is a summary of the author’s PhD thesis supervised by Frits Spieksma and defended on 20 December 2006 at the Katholieke Universiteit Leuven. The thesis is written in English and is available from the author’s website (http://www.econ.kuleuven.be/dries.goossens/public). This work deals with combinatorial auctions, i.e., auctions where bidders can bid on sets of items. We study two special cases, namely the total quantity discount auction and the matrix bid auction.      

A combinatorial proof of a combinatorial theorem

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