Resum In many data mining processes, neighborhood operators play an important role as they are generalizations of equivalence classes which were used in the original rough set model of Pawlak. In this article, we introduce the notion of fuzzy neighborhood system of an object based on a given fuzzy covering, as well as the notion of the fuzzy minimal and maximal descriptions of an object. Moreov...

Resum In many data mining processes, neighborhood operators play an important role as they are generalizations of equivalence classes which were used in the original rough set model of Pawlak. In this article, we introduce the notion of fuzzy neighborhood system of an object based on a given fuzzy covering, as well as the notion of the fuzzy minimal and maximal descriptions of an object. Moreov...

The covering radius R of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning t[ n, k], the minimal covering radius of any binary code of length n and dimension k. For example r[ n, 41 and t [ n, 51 are determined exactly, and reasonably tight bounds on t[ n, k] are obtained for any k when n is large. These results are found by using sev...

Cohen, G.D., S.N. Litsyn, On the covering radius of Reed-Muller codes, Discrete Mathematics 106/107 (1992) 147-155. We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the ‘essence of Reed-Mul...

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optim...

Lahtonen, _I., An optimal polynomial for a covering radius problem, Discrete Mathematics 105 (1992) 313-317. A. Tietlvlinen discovered, how polynomials with suitable Fourier-Krawtchouk coefficients can be used to get information about the covering radius of a code as a function of the dual distance. In this note we determine the optimal polynomials satisfying Tietavlinen’s conditions. This give...

Weanalyse a probabilistic argument that gives a semi-random construction for a permutation code on n symbols with distance n− s and size (s!(log n)1/2), and a bound on the covering radius for sets of permutations in terms of a certain frequency parameter.

Tietäväinen’s upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n2 − ( 2 − o(1)) √ dn and typically at least n2 − Θ( √ dn log nd ). The gap between those bounds on R − n2 is an Θ( √ log nd ) factor related to the gap between the worst covering radius given d and the sphere-covering bound. Our focus in this paper is on the c...

New lower bounds on the minimum length of terror correcting BCH codes with covering radius at most 2t are derived.