Antoine Lobstein

Further results on the covering radius of codes

A number of upper and lower bounds are obtained for K(n, R) , the minimal number of codewords in any binary code of length n and covering radius R . Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply that a linear code is normal. An upper bound is given for the density of a covering code over any alphabet, and it is shown that K(n + 2, R + 1) \leq K(n, R) holds for sufficiently large n .

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Let G=(V,E) be an undirected graph and C a subset of vertices. If the sets Br(v)?C, v?V (respectively, v?V?C), are all nonempty and different, where Br(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively, an r-locating-dominating code). We prove that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.

Identifying and locating-dominating codes on chains and cycles

Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let B r (v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v ∈ V (respectively, v ∈ V\C), the sets Br(v) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles.

The minimum density of an identifying code in the king lattice

Abstract Consider a connected undirected graph G=(V,E) and a subset of vertices C. If for all vertices v∈V, the sets Br(v)∩C are all nonempty and different, where Br(v) denotes the set of all points within distance r from v, then we call C an r-identifying code. For all r, we give the exact value of the best possible density of an r-identifying code in the king lattice, i.e., the infinite two-dimensional square lattice with two diagonals.

Bounds for Codes Identifying Vertices in the Hexagonal Grid

In an undirected graph G=(V,E), a subset

On Identifying Codes in Binary Hamming Spaces

C \subseteq V

On the Density of Identifying Codes in the Square Lattice

is called an identifying code if the sets

Covering Radius 1985-1994

B_1(v) \cap C

The hardness of solving subset sum with preprocessing

consisting of all elements of C within distance one from the vertex v are nonempty and different. We take G to be the infinite hexagonal grid and show that the density of any identifying code is at least 16/39 and that there is an identifying code of density 3/7.

On normal and subnormal q-ary codes

Abstract A binary code C⊆{0,1}n is called r-identifying, if the sets Br(x)∩C, where Br(x) is the set of all vectors within the Hamming distance r from x, are all nonempty and no two are the same. Denote by Mr(n) the minimum possible cardinality of a binary r-identifying code in {0,1}n. We prove that if ρ∈[0,1) is a constant, then limn→∞n−1 log2M⌊ρn⌋(n)=1−H(ρ), where H(x)=−x log2x−(1−x) log2(1−x). We also prove that the problem whether or not a given binary linear code is r-identifying is Π2-complete.