We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex v to be the set of vertices in G that are at distance at most t from v: The vertex v is then said to cover itself and every other vertex in the ball with center v: Our formal problem statement is as follows: Given an undirected graph G and an integer t 1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C: The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C: We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

Abstract-All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [II, k] code. There is also a recent result on the complexity of computing the covering radius. T I.

We derive new conditions for nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented. Keywords: Krawtchouk polynomials, integral roots, perfect codes, switching reconstruction, Radon transform. 3 Research supported by the Guastallo Fellowship and a grant from the Israeli Ministry of Science and Technology. 1. Introduction The binary Krawtchouk polynomial P n k (x) (of degree k) is defined by the following generating function: 1 X k=0 P n k (x)z k = (1 0 z) x (1 + z) n0x : (1) Usually n is fixed, and when it does not lead to confusion it is omitted. The question of existence of integral zeros of Krawtchouk polynomials (or, that is essentially the same, existence of zero coefficients in the expansion of (1 0 z) x (1 + z) n0x ) arises in many combinatorial and coding theory problems. Let us state some of them. 1. Radon transform on Z n 2 [14]. Let f : Z n 2 ! R; then the R...

This paper aims to present a new genetic approach that uses rank distance for solving two known NP-hard problems, and to compare rank distance with other distance measures for strings. The two NP-hard problems we are trying to solve are closest string and closest substring. For each problem we build a genetic algorithm and we describe the genetic operations involved. Both genetic algorithms use a fitness function based on rank distance. We compare our algorithms with other genetic algorithms that use different distance measures, such as Hamming distance or Levenshtein distance, on real DNA sequences. Our experiments show that the genetic algorithms based on rank distance have the best results.