Gerzson Kéri

Covering and radius-covering arrays: Constructions and classification

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For some of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.

On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures

Several known and newly introduced classes of positive reciprocal matrices emerging from pairwise comparisons in multiattribute decision problems are studied in the paper. Mainly qualitative features in connection with consistency and inconsistency are considered in order to extend the range of the available analytical methods regarding pairwise comparisons. By using graph representation of positive reciprocal matrices, graph theoretic approach is applied for the argumentation. The applied notions and the theorems developed in the paper can be useful for eliminating the illogical data that may occur during pairwise comparisons.

Binary Covering Arrays and Existentially Closed Graphs

Binary covering arrays have been extensively studied in many different contexts, but the explicit construction of small binary covering arrays with strength larger than three remains poorly understood. Connections with existentially closed graphs and Hadamard matrices are examined, particularly those arising from the Paley graphs and tournaments. Computational results on arrays generated by column translation, such as the Paley graphs, lead to substantial improvements on known existence results for binary covering arrays of strengths four and five.

The Sherman-Morrison formula for the determinant and its application for optimizing quadratic functions on condition sets given by extreme generators

First a short survey is made of formulas, which deal with either the inverse, or the determinant of perturbed matrices, when a given matrix is modified with a scalar multiple of a dyad or a finite sum of dyads. By applying these formulas, an algorithmic solution will be developed for optimizing general (i. e. nonconcave, nonconvex) quadratic functions on condition sets given by extreme generators. (In other words: the condition set is given by its internal representation.) The main idea of our algorithm is testing copositivity of parametral matrices.

Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t,b,R). In the paper, necessary and sufficient conditions for K(t,b,R)=M are given for M=6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M= =9 for b>=1. For ternary codes, it is shown that K(3t+2,0,2t)=9 for t>=2. New upper bounds obtained include K(3t+4,0,2t)= =2. Thus, we have K(13,0,6)=<36 (instead of 45, the previous best known upper bound).

Bounds for Covering Codes over Large Alphabets

Let Kq(n,R) denote the minimum number of codewords in any q-ary code of length n and covering radius R. We collect lower and upper bounds for Kq(n,R) where 6 ≤ q ≤ 21 and R ≤ 3. For q ≤ 10, we consider lengths n ≤ 10, and for q ≥ 11, we consider n ≤ 8. This extends earlier results, which have been tabulated for 2 ≤ q ≤ 5. We survey known bounds and obtain some new results as well, also for s-surjective codes, which are closely related to covering codes and utilized in some of the constructions.

Combinatorial results on the fitting problems of the multivariate gamma distribution introduced by Prékopa and Szántai

Generalizations of the results of an earlier paper of the second author, related to the problem of fitting a multivarite gamma distribution to empirical data, are discussed in the paper. The multivariate gamma distribution under consideration is the one that was introduced in the paper of Prékopa and Szántai (in Water Resources Research, 14:19–24, 1978), some earlier results on the fitting problem were given in the paper of Szántai (in Alkalmazott Matematikai Lapok 10:35–60, 1984). In the present paper it is proved that the necessary conditions given earlier are not sufficient and some further new, mostly computational results are provided, too. Using the more efficient computation tools we are able now to give the sufficient conditions for dimensions 5 and 6 as well. For higher dimensions we have only necessary conditions and the invention of a suitable necessary and sufficient condition remains an open problem when n is greater than 6. The miscellaneousness of the necessary and sufficient conditions obtained in our new project for n=6 indicates that finding necessary and sufficient conditions in general should be a very hard problem.

A modified stepping-stone algorithm for the transportation problem

The computational efforts required for the stepping-stone algorithm to solve the transportation problem can be reduced by carrying out the transformation of the reduced costs throughout basis changes in the following way. Given a set H as a system of basic cells, let us consider the numbers δij and let be their new values after a cell (i o, j o) ∈ H leaves and another cell (i 1, j 1)∉H enters the basis. We state that where the definitions of the sets C and D are described in both direct and algorithmic way.