Improved lower bound for locating-dominating codes in binary Hamming spaces

Identifying and Locating- Dominating Codes in Binary Hamming Spaces Identifying and Locating-Dominating Codes in Binary Hamming Spaces

Acknowledgements First of all I want to thank my supervisor Professor Iiro Honkala for his continuous support, and patience when long breaks in research took place. Many discussions with Iiro, his collaboration, suggestions for research topics, and careful proofreading made this work possible. I also want to thank Dr. Tero Laihonen for many inspiring discussions, suggestions for research topics...

On Locating-Dominating Codes in Binary Hamming Spaces

Let F = {0, 1} denote the binary field and F the n-dimensional Cartesian product of it. A code is a subset of F. The elements of F (resp. a code) are called words (resp. codewords) and the Hamming distance d(x, y) between two words x, y ∈ F is the number of coordinate positions in which they differ. The Hamming weight w(x) of a word x ∈ F is the number of 1’s in x. The minimum distance of a cod...

Improved bounds on identifying codes in binary Hamming spaces

A sharp lower bound for locating-dominating sets in trees

Let LD(G) denote the minimum cardinality of a locating-dominating set for graph G. If T is a tree of order n with l leaf vertices and s support vertices, then a known lower bound of Blidia, Chellali, Maffray, Moncel and Semri [Australas. J. Combin. 39 (2007), 219–232] is LD(T ) ≥ (n+ 1 + l − s)/3 . In this paper, we show that LD(T ) ≥ (n+ 1 + 2(l − s))/3 and these bounds are sharp. We construct...

Locating-dominating codes in cycles

The smallest cardinality of an r-locating-dominating code in a cycle Cn of length n is denoted by M r (Cn). In this paper, we prove that for any r ≥ 5 and n ≥ nr when nr is large enough (nr = O(r3)) we have n/3 ≤ M r (Cn) ≤ n/3 + 1 if n ≡ 3 (mod 6) and M r (Cn) = n/3 otherwise. Moreover, we determine the exact values of M 3 (Cn) and M 4 (Cn) for all n.