a b s t r a c tBertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locatingdominating codes in paths P n . They conjectured that if r ≥ 2 is a fixed integer, then the smallest cardinality of an r-locating-dominating code in P n , denoted by M LD r (P n ), satisfies M LD r (P n ) = ⌈(n + 1)/3⌉ for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r ≥ 3 we have M LD r (P n ) = ⌈(n + 1)/3⌉ for all n ≥ n r when n r is large enough. In addition, we solve a conjecture on location-domination with segments of even length in the infinite path.

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Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t-neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t-neighbourhoods of the input clues. A varying number of input clues is allowed, but a limit mu on the maximum number of them is imposed. Of course, we would like mu to be as small as possible. We consider the problem over the binary Hamming space F n and focus on the minimum of mu, denoted by ν(n; t). Using linear reference sets, we show that ν(n; 2) ≤ 5 for any n ≥ 9. We also give infinite families of reference sets which provide good bounds on ν(n; t) for t = 3. In addition, efficient methods are given to obtain bounds on ν(n; t) for any t from known reference sets.We also discuss the applications of this model to the Levenshtein's sequence reconstruction problem and sensor network monitoring.

The concept of identifying codes in a graph was introduced by Karpovsky et al. (in IEEE Trans Inf Theory 44 (2):599-611, 1998). These codes have been studied in several types of graphs such as hypercubes, trees, the square grid, the triangular grid, cycles and paths. In this paper, we determine the optimal cardinalities of identifying codes in cycles and paths in the remaining open cases.

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