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...