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Let $G=(V,E)$ be a connected undirected graph and $S$ a subset of vertices. If for all vertices $v \in V$, the sets $B_r(v) \cap S$ are all nonempty and different, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $S$ an $r$-identifying code. We give constructive upper bounds on the best possible density of $r$-identifying codes in four infinite regular graphs, for small values of $r$.

Abstract-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) 5 K(n, R) holds for sufficiently large n.

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

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In an undirected graph G = (V; E) a subset C V is called an identifying code, if the sets B1 (v) \ C consisting of all elements of C within distance one from the vertex v are nonempty and di erent. We take G to be the in nite 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.

A binary code C f0; 1g n is called r-identifying, if the sets B r ðxÞ \ C; where B r ðxÞ is the set of all vectors within the Hamming distance r from x; are all nonempty and no two are the same. Denote by M r ðnÞ the minimum possible cardinality of a binary r-identifying code in f0; 1g n : We prove that if r 2 ½0; 1Þ is a constant, then lim n!1 n À1 log 2 M b rnc ðnÞ ¼ 1 À H ðrÞ; where H ðxÞ ¼ Àx log 2 x À ð1 À xÞ log 2 ð1 À xÞ: We also prove that the problem whether or not a given binary linear code is r-identifying is P 2 -complete. # 2002 Elsevier Science (USA)