Solving Two Conjectures regarding Codes for Location in Circulant Graphs

Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs

On the size of identifying codes in binary hypercubes

We consider identifying codes in binary Hamming spaces F^n, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of F^n. For any subset X of F^n, denote by I_r(X)=I_r(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r,<= l)-identifying code if the sets I_r(X) are distinct for all subsets X of size at most l. We estimate the smallest size of such codes with fixed l and r/n converging to some number rho in (0,1). We further show the existence of such a code of size O(n^{3/2}) for every fixed l and r slightly less than n/2, and give for l=2 an explicit construction of small such codes for r the integer part of n/2-1 (the largest possible value).Comment: 13 page

On identifying codes that are robust against edge changes

AbstractAssume that G=(V, E) is an undirected graph, and C⊆V. For every v∈V, denote Ir(G; v)={u∈C: d(u,v)≤r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets Ir(G; v) for v∈V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r=1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide

On t-revealing codes in binary Hamming spaces

In this paper, we introduce t-revealing codes in the binary Hamming space F-n. Let C subset of F-n be a code and denote by I-t (C; x) the set of elements of C which are within (Hamming) distance t from a word x is an element of F-n. A code C is t-revealing if the majority voting on the coordinates of the words in I-t (C; x) gives unambiguously x. These codes have applications, for instance, to the list decoding problem of the Levenshtein's channel model, where the decoder provides a list based on several different outputs of the channel with the same input, and to the information retrieval problem of the Yaakobi-Bruck model of associative memories. We give t-revealing codes which improve some of the key parameters for these applications compared to earlier code constructions. (C) 2019 Elsevier Inc. All rights reserved

New bounds on binary identifying codes

AbstractThe original motivation for identifying codes comes from fault diagnosis in multiprocessor systems. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.In this paper, we concentrate on identification in binary Hamming spaces. We give a new lower bound on the cardinality of r-identifying codes when r≥2. Moreover, by a computational method, we show that M1(6)=19. It is also shown, using a non-constructive approach, that there exist asymptotically good (r,≤ℓ)-identifying codes for fixed ℓ≥2. In order to construct (r,≤ℓ)-identifying codes, we prove that a direct sum of r codes that are (1,≤ℓ)-identifying is an (r,≤ℓ)-identifying code for ℓ≥2

Optimal local identifying and local locating-dominating codes

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities

On Vertex-Robust Identifying Codes of Level Three

Assume that G = (V, E) is an undirected and connected graph, and consider C subset of V. For every v is an element of V, let I(r)(v) = {u is an element of C : d(u, v) = 2t+1 for any two different vertices u and v. Vertex-robust identifying codes of different levels are examined, in particular, of level 3. We give bounds (sometimes exact values) on the density or cardinality of the codes in binary hypercubes and in some infinite grids

Tolerant location detection in sensor networks

Location detection in sensor networks can be handled with so called identifying codes. For an identifying code to work properly, it is required that no sensors are malfunctioning. Previously, malfunctioning sensors have been typically coped with robust identifying codes. However, they are rather large and, hence, imply high signal interference and energy consumption. To overcome these issues, collections of disjoint identifying codes have been proposed for coping with malfunctioning sensors. However, these collections have some problems regarding detection of malfunctioning sensors and, moreover, it seems unnecessary to restrict oneself to disjoint codes. In this paper, we discuss a certain type of identifying codes, for which the detection of malfunctioning sensors is easy, and based on these codes we design a collection of codes tolerant against malfunctions. We present some results on general graphs as well as optimal constructions in rook's graphs and binary Hamming spaces. (C) 2019 Elsevier Inc. All rights reserved