On Identifying Codes in Binary Hamming Spaces

Abstract

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)

On binary linear r-identifying codes

“…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].…”

Connecting identifying codes and fundamental bounds

“…Identifying codes are also linked to superimposed codes [1,2,[15][16][17], dominating sets [21], locating dominating sets [22], the set cover [19,23] and the test cover problem [19,23] and r-robust identifying codes are linked to error correcting codes with minimum Hamming distance 2r + 1 [10] and the set rmulti-cover problem [23].…”

“…Although introduced only twelve years ago [1], identifying codes have been linked to a number of deeply researched theoretical foundations, including super-imposed codes [2], covering codes [1,3], locating-dominating sets [4], and tilings [5][6][7][8]. They have also been generalized and used for detecting faults or failures in multi-processor systems [1], RF-based localization in harsh environments [9][10][11], and routing in networks [12].…”

Disjoint identifying-codes for arbitrary graphs

“…By coalescing the various reports, the central point can uniquely determine the faulty processor. Several bounds on the minimum size of an identifying code were also presented in [1] and studied further in [7,8] and many subsequent works. In [3,4] identifying codes of radius r = 1…”

“…The closest literature to this work involves the problem of locating-dominating sets [10][11][12]. A set of vertices S is said to be locating-dominating if every vertex not in S is a neighbor of a unique subset of vertices in S. As such, these sets are codes that identify all vertices not in the set, a subtle but important difference from standard identifying codes.…”