We prove the covering radius of third-order Reed-Muller code $\mathrm {RM}(3,7)$ is 20, which was previously known to be between 20 and 23 (inclusive). The maximum nonlinearity among all 7-variable Boolean functions. It that there exi...

Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of codes– the generalized covering radius. The covering-radius hierarchy code characterizes trade-off between storage amount, latency, and access complexity, in systems. Several equivalent definitions are provided, showing this combinatorial, geometric, alg...

In the facility location problem usually reducing total transferring cost and time are common objectives. Designing of a network with hub facilities can improve network efficiency. In this study a new model is presented for P-hub covering location problem. In the p-hub covering problem it is attempted to locate hubs and allocate customers to established hubs while allocated nodes to hubs are in...

D = lim log(.iV(A,c)) (1) E-o log1 where .Af(A, e) denotes the smallest number of closed balls of radius > 0 needed to cover the set A. The aim of this paper is to introduce a new fast algorithm for computing the fractal dimension of binary images according to (1). Consider a binary image that is represented as a matrix of binary pixels stored in a rectangular array I(x, y). Let this image deno...

In this paper we present a family of ternary quasi-perfect BCH codes. These codes are of minimum distance 5 and covering radius 3. The first member of this family is the ternary quadratic-residue code of length 13.

We describe a new invariant that we have used to obtain the complete classification of the cubic forms of nine variables. In particular, we compute the covering radius of RM(2, 9) into RM(3, 9).

We show that all binary codes of lengths 16, 17 and 18, or redundancy 10, are normal. These results have applications in the construction of codes that attain t[n, k], the smallest covering radius of any binary linear code.

We determine the L spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L case, the restriction on the inje...

We discuss parameters of Goppa codes, such as minimum distance, covering radius, distance distribution, and generalized Hamming weights. By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds.

Let (M, g) be a compact Riemannian manifold with universal covering M̃ . The simplest asymptotic invariant which can be associated to (M, g) is the exponential growth rate h(g) of volume on the universal covering, called the volume entropy. If B r (p) denotes the geodesic ball of radius r about p ∈ M̃ and volg(B r (p)) describes its volume with respect to the Riemannian metric g lifted to M̃ then ...