On Binary Codes for Identi®cation

Uri Blass,1 Iiro Honkala,2 Simon Litsyn1

1Department of Electrical Engineering Ð Systems, Tel-Aviv University,Ramat-Aviv 69978, Israel.

2Department of Mathematics, University of Turku, 20014 Turku, Finland

Received May 15, 1999; revised October 7, 1999; accepted October 19, 1999

Abstract: A code C � Fn2 is called t-identifying if the sets Bt�x� \ C are all nonempty and

different. Constructions of t-identifying codes are given. # 2000 John Wiley & Sons, Inc.J Combin

Designs 8: 151±156, 2000

1. INTRODUCTION

Denote by F2 the binary alphabet f0; 1g. The distance d�x; y� between any twovectors x � �x1; . . . ; xn�; y � �y1; . . . ; yn� 2 Fn

2 is the number of coordinates i suchthat xi 6� yi. The number of nonzero coordinates in a vector x is called its weight andis denoted by w�x�. If x; y 2 Fn

2 and d�x; y� � t, then we say that x t-covers y (andvice versa). A nonempty subset C of Fn

2 is called a binary code of length n. Thecovering radius of a code is the smallest integer R with the property that every vectorin Fn

2 is within distance R from at least one codeword. A binary code of length n, Kcodewords and covering radius R is called an �n;K�R code for short.

If C � Fn2 and D � Fn

2 and a 2 Fn2, then we denote

a� C � fa� c j c 2 Cgand

C � D � fc� d j c 2 C; d 2 Dg:As usual, we denote for x 2 Fn

2,

Bt�x� � fy 2 Fn2 j d�y; x� � tg

Uri Blass; E-mail: blass_n@netvision.net.il

Simon Litsyn; E-mail: litsyn@eng.tau.ac.il

Correspondence to: Iiro Honkala; E-mail: honkala@utu.®

# 2000 John Wiley & Sons, Inc.

151

and

St�x� � fy 2 Fn2 j d�y; x� � tg:

De®nition 1. A binary code C of length n is called t-identifying if the setsBt�x� \ C, x 2 Fn

2 are nonempty and different.

These codes were introduced in Karpovsky, Chakrabarty, and Levitin [6]; see also[7]. There is also a closely related problem of ®nding locating±dominating sets, inwhich the requirement is that the sets Bt�x� \ C are nonempty and different for allx 2 Fn

2 n C (i.e., nothing is assumed for x 2 C). For this problem, we refer to the bookby Haynes, Hedetniemi, and Slater [5, Section 7.3].

Assume that 2n processors are arranged in the nodes of an n-dimensional hyper-cube. A processor can check itself and all the neighbors that lie within Hammingdistance t, and reports NO if there is something wrong in the processor itself or one ofthese neighbors and YES otherwise. If we do the checking in the codewords of a t-identifying code and there is something wrong in at most one processor, we know inwhich one, based on the answers we get from the processors corresponding tocodewords.

2. CONSTRUCTIONS

The following theorem is closely related to the construction used in the proof of[6, Theorem 5].

Theorem 1. Let A be a set of binary words of length n and weight t; 1 � t < n=2,with the property that for every 2t-element subset S of the set f1; 2; . . . ; ng there is acollection of words of A such that the union of their supports is S. Let B � Btÿ1�0�:

If D is an �n;K�2t code then the code D� �A [ B� is t-identifying.

Proof. Consider an arbitrary vector w 2 Fn2. Because the covering radius of D is 2t,

there exists a codeword y such that d�w; y� � 2t. We now claim that for all binaryvectors x 2 B2t�y� the sets Bt�x� \ �y� �A [ B�� are nonempty and different; forevery x =2B2t�y� the set Bt�x� \ �y� �A [ B�� is of course empty. This implies thatw can be identi®ed. Denoting x � y� z, where the weight of z is at most 2t, wesee that our claim is the same as requiring that for all z with w�z� � 2t, the setsBt�z� \ �A [ B� are nonempty and different.

Case w�z� � 2t: This is the only case where Bt�z� \ A 6� ; and Bt�z� \ B � ;.By the de®nition of A, the union of the supports of the words a 2 A with d�a; z� � tequals the support of z, and thus identi®es z.

Case t < w�z� � 2t ÿ 1: Among those words of A [ B that have distance at mostt to z the smallest occurring weight is w�z� ÿ t, and this minimum weight identi®esthe weight of z. Moreover, the support of z is the union of the supports of theseminimum weight codewords in Bt�z�.

Case 2 � w�z� � t: Now the minimum weight codeword of A [ B in Bt�z� is theall-zero word. However, the weight of z is identi®ed by the largest weight such thatall words of weight w are in Bt�z�. Clearly, this largest weight is t ÿ w�z�. Moreover,the union of the supports of the words of weight t ÿ w�z� � 1 that are not in Bt�z� isthe complement of the support of z.

152 BLASS, HONKALA AND LITSYN

Case w�z� � 1: In this case all words of weight t ÿ 1 are in Bt�z�. By thede®nition of A, every 2t-element subset not containing the support of z can beobtained as a union of supports of some words in A; hence the union of the supportsof the words in A n Bt�z� is the complement of z, thus identifying z.

Case w�z� � 0: In this case all the words of A [ B belong to Bt�z�. &

Since we can always take A � St�0�, we see that the size of A [ B can always bemade to be at most

Xt

i�0

n

i

� �� nt

t!

when t is ®xed and n!1. The asymptotic effect of B in the previous theorem iszero, which is why we at the moment do not pay any attention to whether thecodewords in B are actually needed. We return to this later.

We can obtain an asymptotical improvement on a result in [6] using the previoustheorem, by considering the remaining `̀ constant weight'' identifying problem.

Construction 1: Consider the coordinate positions 1; 2; . . . ; n modulo 2t ÿ 1,t � 3, and take in A all the words such that not all the elements in the support belongto different residue classes. Then D� �A [ B� is t-identifying, if D and B are as inTheorem 1.

Clearly, the chosen words form a set A with the desired property. Indeed, take anyvector z of weight 2t. In the support of z at least one residue class is represented by atleast two elements, say i and j. Take in turn any t ÿ 2 of the other 2t ÿ 2 elements inthe support to obtain codewords of A whose union gives the support of z.

For instance, for t � 3, we only take (assuming that n is divisible by ®ve)

n

3

� �ÿ 5

3

� �n

5

� �3

elements to A, i.e.,13=25 of the n3

ÿ �vectors of weight 3.

Denote by Mt�n� the smallest cardinality of a t-identifying code of length n. Thenthe density of the smallest t-identifying codes of length n is

��n; t� � Mt�n�2n=V�n; t� :

As usual, we denote by K�n; t� the smallest cardinality of a binary code of length nand covering radius t, and

��n; t� � K�n; t�2n=V�n; t� ;

the density of the smallest binary codes of length n and covering radius t.

BINARY CODES FOR IDENTIFICATION 153

From [6] we know that for a ®xed t and n tending to in®nity

��n; t� � 2t

t

� ���n; 2t��1� o�1��:

We can obtain an improvement which is exponential in t:

Construction 2: We take in A all the t-tuples such that at least t ÿ 1 of them belongto the same residue class modulo 2. Again, this is clearly a valid choice satisfyingthe condition in Theorem 1, and D� �A [ B� is t-identifying if D and B are as inTheorem 1.

Theorem 2. For a ®xed t and n tending to in®nity

��n; t� � t � 1

2tÿ1

2t

t

� ���n; 2t��1� o�1��:

Proof. The cardinality of A is at most

2dn=2et ÿ 1

� �dn=2e � 2

dn=2et

� �:

The result now follows from Theorem 1. &

Now we return to the topic discussed after Theorem 1, namely that if A is suitablychosen, we may be able to take a smaller B. In the following case we choose B � ;.Theorem 1 of course no longer applies, but a similar argument goes through.

Construction 3: Let t � 2 and A consist of all vectors of weight two except thevectors whose supports are fi; i� 1g for i � 1; 2; . . . ; nÿ 1 and fn; 1g. If D is abinary code of length n and covering radius 4, then D� A is 2-identifying.

The next proof shows the validity of the construction.

Theorem 3. For all n � 7,

M2�n� � nÿ 1

2

� �K�n; 4�:

Proof. Let y 2 Fn2 be arbitrary. There is a codeword c 2 D such that y 2 B4�c�.

Without loss of generality, assume that c � 0. We show that already A identi®es y.For all z 2 Fn

2 of weight at least ®ve, the set B2�z� \ A is empty. It suf®ces to showthat the sets B2�z� \ A are nonempty and different for the vectors z 2 B4�0�.

If z � 0, then B2�z� \ A � A, and this is clearly the only case when this is true.Assume that z 6� 0.

The union of the supports of the words in B2�z� \ A contains more than fourelements if and only if w�z� � 1 or w�z� � 2. If w�z� � 2, then there are exactly twoelements in this union that appear at least four times each in the supports of the wordsin B2�z� \ A whereas the other elements appear at most twice. These two elementsclearly form the support of z. If w�z� � 1, then there is a unique element in this union

154 BLASS, HONKALA AND LITSYN

that appears at least four times in the supports of the words in B2�z� \ A, and it is theunique element in the support of z.

It, therefore, suf®ces to consider the cases w�z� � 3 and w�z� � 4. In the lattercase, the union of the supports of the words in B2�z� \ A contains four elements,namely the elements of the support of z. In the former case, this union contains onlytwo or three elements. If there are three elements in this union, then we immediatelyknow the support of z. Moreover, there can be only two elements only in the casewhen the support of z consists of three consecutive elements, and then the uniqueword in B2�z� \ A is the one obtained by changing the middle 1 to 0. So, in all caseswe can identify z. &

This leads to the following numerical improvement. In the parenthesis we alwaysgive the best previous upper bound from [6].

Corollary 4. M2�20� � 87552 (90000). &

The following result is obvious.

Theorem 5. Mt�t � 1� � 2t�1 ÿ 1 when t � 1. &

Theorem 6. M2�4� � 6 �9�,M2�5� � 6 �12�,M2�6� � 8 �16�.

Proof. M2�4� � 6: Take as codewords 0000, 0001, 0010,0101, 1010, 1100. To provethe lower bound, assume that we have an identifying code with ®ve codewords.Without loss of generality, some three of them are of even weight and it is easy to seethat there are ®ve vectors in the space that are covered by all of them, and we cannotidentify them with two codewords.

M2�5� � 6: Take as codewords 10100, 01010, 00101, 10010, 01001, 11111. Thelower bound is from [6].

M2�6� � 8: Take as codewords 000011, 000101, 001110, 010101, 011000,100010, 101001, 111111. The lower bound is from [6]. &

Theorem 6 has been proved independently by G. Exoo [4].

3. A RELATED IDENTIFICATION PROBLEM

In view of the practical motivation discussed in the Introduction, it is necessary torequire that none of the sets Bt�x� \ C is nonempty (because this corresponds to thecase when none of the processors is malfunctioning). From a mathematical point ofview it would also be natural to consider the problem where (in De®nition 1) we onlyrequire that the sets Bt�x� \ C are all different, i.e., one of these sets could be empty.Denote by Mt�n� the minimum number of codewords in such a code.

Theorem 7. Mt�n� � Mt�n� � 1�Mt�n�.Proof. This is trivial: if C attains the bound Mt�n�, then there is at most one vector xsuch that Bt�x� is empty, and by taking x as a codeword we obtain a t-identifyingcode. &

BINARY CODES FOR IDENTIFICATION 155

Theorem 8. Mnÿ1ÿt�n� � Mt�n�.Proof. Assume that C attains the bound Mt�n�. Since we are dealing with binaryvectors, we know that d�x; c� � nÿ d�x; 1� c� for all x; c 2 Fn

2, and thereforeBnÿ1ÿt�x� \ C � C n Bt�1� x�. Since we know that all the sets Bt�y� are different, soare all the sets C n Bt�1� x�. &

By the previous theorem it suf®ces to study the values Mt�n� for n � 2t � 1. WeconjectureÐbut cannot proveÐthat

Mt�n� � Mt�n� 1� for all n � 2t � 1

and that the same is true for the function Mt�n�.The following immediate corollary records a simple but nontrivial case in which

Mt�n� 6� Mt�n�.

Corollary 9. M1�4� � 7, M1�4� � 6.

Proof. The ®rst equation has been proved in [1]. By Theorem 7, M1�4� �M1�4� ÿ 1 � 6. By Theorems 8, 7 and 6 we have M1�4� � M2�4� � M2�4� � 6. &

ACKNOWLEDGMENT

The authors would like to thank the referees for the detailed reports and their usefuland constructive comments.

REFERENCES

[1] U. Blass, I. Honkala, and S. Litsyn, Bounds on identifying codes, Discrete Math., (toappear).

[2] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering codes. Elsevier, Amsterdam,1997.

[3] G. Cohen, I. Honkala, A. Lobstein, and G. ZeÂmor, New bounds for codes identifyingvertices in graphs, Electronic J. Combinatorics, 6(1) (1999), R19.

[4] G. Exoo, Computational results on identifying t-codes, in preparation.

[5] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Domination in graphs. MarcelDekker, 1998.

[6] M. G. Karpovsky, K. Chakrabarty, and L. B. Levitin, On a new class of codes foridentifying vertices in graphs, IEEE Trans. Inform. Theory, 44 (1998), 599±611.

[7] M. G. Karpovsky, K. Chakrabarty, L. B. Levitin, and D. R. Avresky, On the covering ofvertices for fault diagnosis in hypercubes. Inform. Processing Lett., 69 (1999), 99±103.

156 BLASS, HONKALA AND LITSYN