We will be considering sets of n-tuples over an alphabet A, in two important cases: A ¡ ¢ 0£ 1¤ (binary code); n¤ , all entries of each word distinct (set of permutations). We often impose closure conditions on these sets, as follows: A binary code is linear if it is closed under coordinatewise addition mod 2. A set of permutations is a group if it is closed under composition. x£ yïs the number...

Sol& P., A. Ghafoor and S.A. Sheikh, The covering radius of Hadamard codes in odd graphs, Discrete Applied Mathe-atics 37/38 (1992) 501-5 10. The use of odd graphs has been proposed as fault-tolerant interconnection networks. The following problem originated in their design: what is the graphical covering radius of an Hadamard code of length 2k1 and siLe 2k1 in the odd graph Ok? Of particular i...

The purpose of this paper is to discuss some recent generalizations of the basic covering radius problem in coding theory. In particular we discuss multiple coverings, multiple coverings of the farthest-o points and weighted coverings.

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R = 2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r = 2k + 1 (the case q = 3, r = 4k + 1 was considered ...

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Recently the authors gave a new proof of a classical lower bound of Rodemich on Kq(n, n−2) by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich’s original proof, the method generalizes to lower-bound Kq(n, n − k) for any k > 2. The approach is bes...

We simplify the proofs of four results in [3], restating two of them for greater clarity. The main purpose of this note is to give a brief transparent proof of Theorem 7 of [3], the main upper bound of that paper. The secondary purpose is to give a more direct statement and proof of the integer programming determination of covering radius of [3]. Theorem 7 of [3] follows from a simple result in...

Given a set P of n points in the plane and a multiset W of k weights with k ≤ n, we assign each weight in W to a distinct point in P to minimize the maximum weighted distance from the weighted center of P to any point in P . In this paper, we give two algorithms which take O(kn log n) time and O(kn log k + kn log n) time, respectively. For a constant k, the second algorithm takes only O(n log n...

Some of the principal unsolved problems related to the covering radius of codes aredescribed. For example, although it is almost twenty years since it was built, ElwynBerlekamp’s light-bulb game is still unsolved. [Added later: this problem has since beensolved — see P. C. Fishburn and N. J. A. Sloane, ‘‘The Solution to Berlekamp’sSwitching Game,’’ Discrete Math., Vol. 74, 1989,...

Remark 4: The author is unable to deal analytically with the general case of p + l/2 where one does not have the property of symmetry. However, the case that p is close to l/2 may be tractabie an9 interesting. Linear smoothing using measurements containing correlated noise with an application to inertial navigation, " IEEE Trans. Abstract-The covering radius is given for all binary cyclic codes...

In this work a heuristic algorithm for obtaining lower bounds on the covering radius of a linear code is developed. Using this algorithm the least covering radii of the binary linear codes of dimension 6 are determined. Upper bounds for the least covering radii of binary linear codes of dimensions 8 and 9 are derived.