Constructions of nonlinear covering codes are given. Using any nonlinear starting code of covering radius R 2 these constructions form an infinite family of codes with the same covering radius. A nonlinear code is treated as a union of cosets of a linear code. New infinite families of nonlinear covering codes are obtained. Concepts of R; l-objects, R; l-partitions, and R; l-length are described...

We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (Mn, g) is a complete Riemannian manifold with filling radius at least R, then it contains a ball of radius R and volume at least δ(n)Rn . If (Mn, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most δ(n)V ol(M,hyp), then the universal cover of (M, g) contains a unit ball with vol...

Let R(r, m) by the rth order Reed-Muller code of length 2 )n, and let o(r, m) be its covering radius. We obtain the following new results on the covering radius of R(r, m): 1. p(r + 1, m + 2) > 2p(r, m) + 2 if 0 < r < m 2. This improves tile successive use of the known inequalities p(r + 1, m + 2) _> 20 (r + 1, m + 1) and p (r + 1, m + 1) -> O (r, m). 2. P (2, 7) -< 44. Previously best known up...

We introduce a new approach which facilitates the calculation of the covering radius of a binary linear code. It is based on determining the normalized covering radius p. For codes of fixed dimension we give upper and lower bounds on p that arc reasonably close. As an application, an explicit formula is given for the covering radius of an arbitrary code of dimension <4. This approach also sheds...

We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tiett avv ainen 10] and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The new upper bound on the information rate is an...

Bhandari, M.C. and M.S. Garg, A note on the covering radius of optimum codes, Discrete Applied Mathematics 33 (1991) 3-9. This paper gives a lower bound and an upper bound for the covering radius of optimum codes. The upper bound so obtained is better than other known upper bounds, restricted to optimum codes. Optimum codes of covering radius d1 and d2 are shown to be normal. A binary linear co...

Let Rt,n denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary ReedMuller code RM(2, 6) in the sets Rt,6, t = 0, 1, 2 is 16. Second, we show that the covering radius of the binary Reed-Muller code RM(2, 7) in the set R3,7 is 32. We derive a new lower bound for the covering radius of the Reed-Muller code RM(2, n) in the set Rn−4,n....

Fuzzy covering-based rough set models are hybrid models using both rough set and fuzzy set theory. The former is often used to deal with uncertain and incomplete information, while the latter is used to describe vague concepts. The study of fuzzy rough set models has provided very good tools for machine learning algorithms such as feature and instance selection. In this article, we discuss diff...

In [5], we studied binary codes with covering radius one via their characteristic functions. This gave us an easy way of obtaining congruence properties and of deriving interesting linear inequalities. In this paper we extend this approach to ternary covering codes. We improve on lower bounds for ternary 1-covering codes, the so-called football pool problem, when 3 does not divide n − 1. We als...