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 paper, the performance of a new fuzzy classifier, here called fuzzy β-NN, has been analyzed. The classifier classifies data according to the fuzzy membership values of the reference set inside the prespecified radius β. Members of the reference set outside the radius β have no influence on classification decision. The successful classification by the classifier depends on the parameters...

In case of fuzzy reasoning in sparse fuzzy rule bases, the question of selecting the suitable fuzzy similarity measure is essential. The rule antecedents of the sparse fuzzy rule bases are not fully covering the input universe therefore fuzzy reasoning methods applied for sparse fuzzy rule bases requires similarity measures able to distinguish the similarity of non-overlapping fuzzy sets, too. ...

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.

We prove that the smallest covering code of length 8 and covering radius 2 has exactly 12 words. The proof is based on partial classi cation of even weight codewords, followed by a search for small sets of odd codewords covering the part of the space that has not been covered by the even subcode.

the present study was conducted to determine peak ground acceleration (pga) over bedrock in probabilistic analysis methods for the seismic hazard and uniform hazard spectra at different hazard levels for esfahan city. a series of statistics containing historical and instrumental seismic data covering from the 8th century a.d. to the now up to a radius of 200 km was employed and seismic sources ...