Improved bounds on identifying codes in binary Hamming spaces

Exoo, Geoffrey; Junnila, Ville; Laihonen, Tero; Ranto, Sanna

doi:10.1016/j.ejc.2009.09.002

Cited by 18 publications

(14 citation statements)

References 15 publications

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“…By Lemma 4 we can improve the lower bound from [8] and [4] for linear codes, see the proof in [4]. We get the bound in Theorem 8 which improves some values compared with Theorem 5.…”

Section: Lower Boundssupporting

confidence: 47%

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On binary linear r-identifying codes

Ranto

2010

Des. Codes Cryptogr.

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A subspace C of the binary Hamming space F n of length n is called a linear r -identifying code if for all vectors of F n the intersections of C and closed r -radius neighbourhoods are nonempty and different. In this paper, we give lower bounds for such linear codes. For radius r = 2, we give some general constructions. We give many (optimal) constructions which were found by a computer search. New constructions improve some previously known upper bounds for r -identifying codes in the case where linearity is not assumed.

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“…By Lemma 4 we can improve the lower bound from [8] and [4] for linear codes, see the proof in [4]. We get the bound in Theorem 8 which improves some values compared with Theorem 5.…”

Section: Lower Boundssupporting

confidence: 47%

“…The cardinality of an optimal r -identifying code of length n is denoted by M r (n). The value of M r (n) is considered in several papers, see for example [1][2][3][4][5][6][7][8]12].…”

mentioning

confidence: 99%

On binary linear r-identifying codes

Ranto

2010

Des. Codes Cryptogr.

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“…The sets A α depend on γ , but by selecting a subsequence, we may assume that for each α ∈ F , either |A α | = a α for some finite a α (9) or |A α | → ∞.…”

Section: Theorem 8 Letmentioning

confidence: 99%

On the size of identifying codes in binary hypercubes

Janson

Laihonen

2009

Journal of Combinatorial Theory, Series A

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In this paper, we consider identifying codes in binary Hamming spaces F n , i.e., in binary hypercubes. The concept of (r, )-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 us denote by M ( ) r (n) the smallest possible cardinality of an (r, )-identifying code in F n . In 2002, Honkala and Lobstein showed for = 1 that lim n→∞ 1 n log 2 M ( ) r (n) = 1 − h(ρ),where r = ρn , ρ ∈ [0, 1) and h(x) is the binary entropy function. In this paper, we prove that this result holds for any fixed 1 when ρ ∈ [0, 1/2). We also show that M ( )for every fixed and r slightly less than n/2, and give an explicit construction of small (r, 2)-identifying codes for r = n/2 − 1.

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“…If c is an element of Q n and r is a non-negative integer, then B(c, r) denotes the ball of center c and radius r, that is, B(c, r) := {v : v ∈ Q n , d(c, v) ≤ r}. The most recent upper and lower bounds on i (l) r (Q n ) were proved in [5,10,8,9,14]. Adaptive identification in Q n was studied by Junnila [15] who obtained lower and upper bounds on a (1) 1 (Q n ).…”

Section: Introductionmentioning

confidence: 96%

Identifying codes and searching with balls in graphs

Kim

Kumbhat

Nagy

et al. 2015

Discrete Applied Mathematics

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a b s t r a c tGiven a graph G and a positive integer R we address the following combinatorial search theoretic problem: What is the minimum number of queries of the form ''does an unknown vertex v ∈ V (G) belong to the ball of radius r around u?'' with u ∈ V (G) and r ≤ R that is needed to determine v. We consider both the adaptive case when the jth query might depend on the answers to the previous queries and the non-adaptive case when all queries must be made at once. We obtain bounds on the minimum number of queries for hypercubes, the Erdős-Rényi random graphs and graphs of bounded maximum degree.

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Improved bounds on identifying codes in binary Hamming spaces

Cited by 18 publications

References 15 publications

On binary linear r-identifying codes

On binary linear r-identifying codes

On the size of identifying codes in binary hypercubes

Identifying codes and searching with balls in graphs

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