Optical Orthogonal Codes With Unequal Auto-correlation and Cross-correlation Constraints

8/9/2019 Optical Orthogonal Codes With Unequal Auto-correlation and Cross-correlation Constraints

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Optical Orthogonal Codes with UnequalAuto- and Cross-Correlation Constraints

Guu-Chang Yang, Member, IEEE, and Thomas E. Fuja, Member, IEEE

Abstract- An optical orthogonal code (OOC) is a collectionof binary sequences with good auto- and cross-correlation prop-erties; they were defined by Salehi and others as a means ofobtaining code-division multiple access on optical networks. Up tonow, all work on OOCs have assumed that the constraint placedon the autocorrelation and that placed on the cross-correlationare the same. In this paper we consider codes for which the twoconstraints are not equal. Specifically, we develop bounds on thesize of such OOCs and demonstrate construction techniques forbuilding them. The results demonstrate that a significant increasein the code size is possible by letting the autocorrelation constraintexceed the cross-correlation constraint. These results suggest thatfor a given performance requirement the optimal OOC may beone with unequal constraints.This paper also views OOCs with unequal auto- and cross-correlation constraints as constant-weight unequal error pro-tection (UEP) codes with two levels of protection. The boundsderived are interpreted from this viewpoint.

Index Terms-Code-division multiple access, optical networks,constant-weight codes, unequal error protection codes, spread-spectrum systems.I. INTRODUCTION

HIS PAPER concerns the use of optical fiber in a mul-T iuser communication network. Specifically, it considersthe use of code-division multiple-access techniques that permitmany users to share a single optical channel through theassignment of unique signature sequences.This approach has a long history as applied to communi-cation channels where the modulated signals can have bothpositive and negative components-e.g., binary phase-shiftkeying. However, in optical systems-where incoherent pro-cessing means that only signal intensity is measured-thereare no negative components, and the effect on code design isprofound. This was noted by Salehi and others in their designof optical orthogonal codes (OOCS) [1]-[4]The results in this paper extend previous work on opticalorthogonal codes in that we consider codes for which theauto- and cross-correlation constraints are not equal. Weobserve that the effects of the two constraints on system

Manuscript received August 28, 1992; revised March I , 1994. T. Fujawas supported in part by the National Science Foundation under GrantNCR-8957623 and by the NSF Engineering Research Center Program CD R-8803012. Portions of this work were presented at the 1991 IntemationalSymposium on Comm unications, Tainan, Taiwan, December 1991, and atthe 1992 Conference on Information Sciences and Systems, Princeton, NJ,March 1992.G.-C. Yang is with the Department of Electrical Eng ineerin g, NationalChung-Hsing University, Taichung, Taiwan, ROC.T. Fuja is with the Electrical Engineering Department, Institute for SystemsResearch, University of Maryland, College Park, MD 20742 USA.IEEE Log Number 9406409.

performance are not the same, and so considering only codesfor which the constraints are identical may lead to a suboptimalcode. Bounds on such OOCs are derived and techniques forconstructing them are described.Finally, OOCs with unequal auto- and cross-correlationconstraints may be viewed as constant-weight unequal errorprotection (UEP) codes; therefore, we interpret the bounds andconstructions in that context.11. BACKGROUNDND MOTIVATION

In this section we briefly review previous work on opticalorthogonal codes and indicate why the problem considered inthis paper is important.A. Definitions and Past Work

What follows is the definition of an OOC given by Salehiet al . [3].Definition: An (n, w, A A optical orthogonal code C sa collection of binary n-tuples, each of Hamming weight w,such that the following two properties hold:

For any z = [zo, . . ,xn-l] E C andAutocorrelation)any integer r,O < r < n,n- 1

t= O

(Cross-correlation) For anyz = [zo, . . . 5 , -1 ] E C ndany y = [ y ~ , . . . , y ~ - ~ ]C such that z # y and anyinteger r

n- 1C X t Y t H , , L At= O

Note: OOCs were defined in terms of periodic corre-lation; thus the addition in the subscripts above-denoted@-is all modulo-n.The definition of an OOC can be recast in terms of Ham-ming distance; doing so makes clearer the parallels betweenOOCs and constant weight error correcting codes.

Notation: Given a binary n-tuple z, et Dix denote thebinary n-tuple obtained by performing i right-cyclic shifts onAlternate Definition: An ( n ,w, A A optical orthogonalcode C s a collection of binary n-tuples, each of Hammingweight w, such that the following two properties hold:

(Autocorrelation) For any z C, gin(z) w - Awhere dgin(z ) s the minimum distance between z nd

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YANG AND FUJA: OPTICAL ORTHOGONAL CODES 91

Aits cyclic shifts-i.e., d,;, (2) min { d ~ ( x ,x ) : =(Crosscorrelation) For any x E C and anyy E C,dLin(2, ) 2 2w - 2A,, whered L l n ( x ,y) = min{dH(x, DY):T = 0, I , . . . , n - 1).

Consider the partition of binary n-tuples into clouds,where every cloud consists of cyclic shifts of the same n-tuple. Then constructing an OOC consists of picking at mostone n-tuple from every cloud under two constraints; the firstconstraint specifies the minimum Hamming distance withina cloud, while the second specifies the minimum Hammingdistance between clouds. Thus if the two constraints areequal-i.e, if A = A = A-then an OOC, taken togetherwith all of the cyclic shifts of each OOC codeword, representsa constant-weight cyclic error-correcting code with minimumdistance 2w - 2A.The use of OOCs for multiple access is described in[I], [2]. We assume that each user may transmit a physicaloptical pulse-a chip-at any time. If we assume incoherentprocessing-i.e., only the intensity and not the phase ofthe signal is available at the receiver-then multiple pulsestransmitted simultaneously by different users sum.

Each user in the network is assigned a codeword from anoptical orthogonal code. The codeword assigned to a user isthat users signature sequence, and when the user wishes toconvey a logical 1 he transmits the corresponding sequenceof pulses and pauses; when the user wishes to convey a logical0 he transmits nothing for n chip durations. At the receiver,each user computes the correlation of the received sequencewith that users signature sequence; because of the low auto-and cross-correlation properties the correlation typically stayslow untl a logical 1 is indicated by a correlation of w . Inthis way each user can recover his own logical sequence.In [ I ] , [ 2 ] Salehi introduced optical orthogonal codes andcomputed the error probability assuming a channel where theonly noise is interference from other users. In [3] Chung,Salehi, and Wei described constructions of OOCs for the caseA = A = 1 and derived bounds on the cardinality on an( n ,w , A, A) code. In [4J Chung and Kumar described a con-struction technique for the case A = A = 2 and derived newupper bounds on the cardinality of an optimal OOC-again forthe case A = A = A. Finally, we note the work of A, Gyorfi,and Massey [ 5 ]on cyclically permutable codes (CPCs)-i.e.,binary codes whose codewords are cyclically distinct withfull cyclic order; a constant weight CPC is exactly an opticalorthogonal code.

1,2 , . . . , n - I}.

A

B. Why Consider A # A c ?The two correlation constraints serve two purposes.

The autocorrelation constraint guarantees that each signa-ture sequence is unlike cyclic shifts of itsel f. This propertyenables the receiver to obtain synchronization-to findthe beginning of its message and locate the codewordboundaries.The cross-correlation constraint guarantees that each sig-nature sequence is unlike cyclic shifts of the other sig-nature sequences. This property enables the receiver to

estimate its message in the presence of interference fromother users. Thus the cross-correlation constraint aids syn-chronization in the presence of multiple users and permitseach receiver to track its message after synchronizationis achieved.Thus the autocorrelation constraint contributes only to syn-chronization, while the cross-correlation constraint affects both

synchronization and operation.A reasonable figure of merit for a code is the numberof interfering users necessary to cause the code to fail. Forinstance, assume synchronization has been achieved; thenthe only errors the ith receiver can make in estimating itslogical sequence are 0 4 1 errors, and they can only occurwhen enough other users interfere to make the correlation atthe ith receiver exceed a threshold w . Since each of thoseother users can contribute at most A to the correlation, theperformance figure of merit is w/A,. In a similar vein,the synchronization figure of merit is (w - A,)/& formultiple-access synchronization and w - A for single-usersynchronization.Taking these as our performance criteria, we see whyasymmetric OOCs-i.e., codes with A # &-might bepreferable to symmetric codes. If we compare (for instance)an (n ,w + m , A+m, A) OOC with either an ( n ,w , A, A) codeor an ( n ,w +m , A +m, A +m ) code, we see the asymmetriccode is more robust.

So the performance of an ( n ,w +m , A + m, A) OO C is atleast as good as comparable symmetric OOCs. However,we will demonstrate that the cardinality of the (n ,w +m, A +m , A ) code can actually exceed that of the less robust codes;thus more users can be provided even better performance. Thismotivates the study of such OOCs.111. S O M E NEW BOUNDS N OPTICAL ORTHOGONAL CODES

Define @ (n , , A A to be the cardinality of an optimaloptical orthogonal code with the given parameters, i.e.,A@ (n , , A A = max{(C(: C is an (n ,w , A A OOC}.

In this section we derive some new bounds on @ ( n , ,A,). Before this can be done, however, we need to set up thenotation and derive some preliminary results.Dejnition: Let x = [xo, 1 , . . ,x,-I] be a binary n-tuple

xjw-l = 1. Thef weight w ; assume xj o = xjl = . . . -adjacent relative delay vector associated with x is denotedt , = [ t o , t l , . . . , t w - l ] and is defined by

-

f o r i = O , 1 , . . . , w - 2for i = w - 1.+ o- jlu-l,More generally, the relative delay between two 1s in a binaryn-tuple is the number (modulo-n) of cyclic shifts required toline up the two 1s;tz consists of the relative delay betweenall adjacent 1s in 2.Notation: Let x be a binary n-tuple of weight w and lett x = [ t o , t l , . . , w-l] be its adjacent relative delay vector.Let Rx = [ rZ( i , ) ] denote the (w - 1) x w array of integers



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whose (2, j)th element is given by

(Note: The subscript addition above and in the definition ofMz ,

For any z (0, 1) and any integer X ( 1 5 X 5 w - 1) letM z ,below is all modulo-w-denoted U.)be the set of integer A-tuples given by

i r io iz

i , -1 1

0 5 io < il < . . . < ix-1 5 w 2 ,j = 0 , l , . . . , W - l 1

where tz = [ t o , t l , . . . , t w - l ] .There are at most w(,) vectors in Mz,x; there are(l)ays to pick the i l s and w ways to pick the js. Ifevery such selection yields a different vector then IMz, X I =w ( ~ ; ) ; therwise IMz,xl < w(;).Example: Let z = [1001100010000]. Then

tz = [ t o , t i , t 2 , t3 ] = [3, 1, 4, 51.Furthermore

and

n - 1C:I:t3:tCBrt=O

holds for all 1 5 T 5 n - 1 if and only if no component ofRz appears more than X times.

Proof: The elements of Rz indicate the relative delaybetween every pair of 1s in z.Therefore, Rz containsX + 1 repeated elements if and only there exist two sequences{io, i l , . . . , Z ~ }nd {i;, zi,...,zL} such that for all j =0, l , . . . , X

xa 3= xa l = 1 and i, - : = I-* # 0.3But this is true if and only if xtzt+.r*2 X + 1.Q.E.D.

L em m a 2 : Let z = [ z 0 , ~ l , . . . , x , - 1 ] and y =[yo, y1, . . ,y n - l ] be binary n-tuples. Then the inequality

t= Oholds for all 0 5 7 5 n - 1 if and only if M z ,are disjoint. is a collection of integer A-tuples. A vec-tor m = [ao, a l ; . . , a x - l ] is in Mz,x if and only ifthere exists a sequence of X + 1 distinct integers--call themio , l , . ., x-such that

xa J= 1 , f o r j = 0 , l , - . . , X

and My,P r o o j M z ,

andi 3 + 1 - i 3 = a , , f o r j = O , l , . . . , X - l .

Therefore, M z, nMy, = 0 if and only if it is impossible toline up X + 1binary 1s in zwith X + 1binary 1s in y withcyclic shifts-i.e., if and only if x ty tB 7 5 A. Q.E.D.Lemma 3: Let z= [ZO, 5 1 , . . . ,z,-1] be a binary n-tuple.Then the inequality

t = Oholds for all T = 1, Z, . . . , n - 1 if and only if IMz,xl =w(xl)-i .e., if and only if the vectors defining M z , x areall distinct.Proof: Similar to that of Lemma 2 . Each A-tuple inMz, corresponds to X + 1 nonzero components of E suchthat the relative delays between the nonzero components aregiven by the A-tuple. If lMz , < w w;l) then there aretwo different sets of X + 1 nonzero components with thesame relative delays between them; thus we can line upthe X + 1 binary ones and obtain an autocorrelation of at leastX + 1. Conversely, if IMz,X I = w ( ;I) then every set ofX + 1 nonzero components of z ave a different relative delaystructure, so it is impossible to obtain an autocorrelation ofX + 1 or more. Q.E.D.A. An Upper Bound

In this section we use the lemmas above to provide an upperbound on @ ( n , , Xc ) .First consider the case Xa = Xc = A; the bound we deriveis identical to one in [3 ] derived from the Johnson bound forconstant weight error-correcting codes; it is rederived here toillustrate the approach that will be taken in the proof of thenew bounds.



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Theorem 1: [Johnson Bound] The following inequality We note also that Theorem 2 is only a generalization ofholds: the Johnson bound for A = 1; for A 2 2 the bound on@(n,w, A, A) obtained by setting m = 0 in Theorem 2 isweaker than the bound in Theorem 1.( n- I)(. - 2 ) . . (n - A )w(w - 1). .(w - A) .(n , , A, A) IProof: Let C be an optimal (n ,w, A, A ) OOC-i.e.,IC) = @(n,w, A, A) . From Lemma 3 we know that for every

z E C the set M z , consists of w( ") distinct integer A-tuples. Furthermore, from Lemma 2 we know that for z, EC, z # y, the sets Mz, and My, are disjoint. Therefore,the union of M z , x as z aries over all z E C consists of@(n, w, A, A ) .w (w;l) distinct integer A-tuples. However, if[ao, l,...,ax-1]M z , x thenao+al+. . .+ax- l 5 n-1.The number of ways to select A positive ai's that sum to nomore than n - 1 is just the number of compositions of n withA + 1 positive parts-and that is equal to ( We havethus shown that

q n , w , A, A ) . w ('";I) I (y )which was to be proven. Q.E.D.Our next goal is to bound @(n,w, A A for A > A Todo so we first need a preliminary lemma.Lemma 4: Let z E C, where C is an ( n, w, A + m , A )optical orthogonal code. (Assume m 2 0 is an integer.) Then

U(" x ')A + m .Mz,xl 2

Proof: See Appendix I.Theorem 2: Let m be a nonnegative integer. Then

( n- 1)(n- 2 ) . . .(n - A) (A + m )w(w - 1)(w - ) . . (w - A) .(n,w, A + m , A) I

Proofi Let C be an ( n, w, A+m, A) OOC such that (CI@(n,w, A + m, A) . By Lemma 4, or any z E C , IMz,xJ2w( "i')/(m+A). Furthermore, by Lemma 2 Mz, and My,are disjoint forz, E C andz # y; therefore, IMz, I summedup over all hz E C cannot exceed the total number of integerA-tuples that are "allowable" as elements of M E ,Aa numbershown in the proof of Theorem 1 to be ( Hence

-(" x l )2 @(n,w, A + m, A ) . m + Awhich was to be proven. Q.E.D.Examining Theorem 2, we find (for instance) that the upperbound on @(n,w, A, 1) is A times greater than the analogousbound on @(n,w, 1, 1). Note also that a trivial upper boundon @(n,w, A + m , A) is given by any upper bound on@(n,w , A + m , A + m ) . For "typical" OOC values-i.e.,n >> w-an upper bound derived this way will be looser thanthe bound in Theorem 2. For instance, considering (n, w, 2 , 1)OOC's, the bound in Theorem 2 is tighter than the Johnsonbound for ( n ,w, 2 , 2) codes provided n > 2w - 2 .

B. Lower BoundsIn [3], [6] a lower bound on @(n,w, A A was derived

for odd prime n . Subsequently, Wei [7] derived an alternatelower bound-again for odd prime n. In what follows we usethe general approach of Wei [7] to bound @ ( n , , A A,) forA # A and any n.

Theorem 3:

where

if x = 0;a(x) = { otherwiseand

Proof: As in [3], [6], the proof consists of demonstratingthat A is an upper bound on the number of binary n-tuplesthat violate the autocorrelation constraint and B is an upperbo.und on the number of binary n-tuples that violate the cross-correlation constraint for a given binary n-tuple z. he resultfollows from an application of the greedy algorithm. Thevalidity of B as an upper bound was demonstrated in [31, [61.Thus the proof consists of demonstrating that there are at mostA binary n-tuples that violate the autocorrelation constraint.A proof of this is given in Appendix 11.C. Asymptotic Bounds

In this section we examine how the cardinality of an opti-mal ( n, w, A A optical orthogonal code behaves for largeblocklength. The goal is to see how quickly the parameters w,A and A should grow the blocklength n in an (n ,w, A Aooc.Lemma 5: Let A be a positive integer, and let p and q be anonnegative constants such that p > (A, + q ) / ( A , + 1) .Then

n-00im @ ( n , anP1, [pnq], A = Ofor any positive real a and p.



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( n- 1 ) . . n- A,)(pnq + 1)c y n p ( c y n p - 1) . . . c y np -

Since by assumption A + q - p ( A , + 1) < 0 we have thedesired result. Q.E.D.Lemma 6: Let A and A be positive integers, and let p be apositive constant such that p < min (A,/(2A, +3) , A,/(2AC+

3 ) ) . Thenlim @(n , anp ] , A = ccn-03

for any positive real a .Proof: See Appendix 111.Consider the case A = A = 1. Lemma 5 indicates that

if the codeword weight grows faster than ,% there are nocodewords for large n. Lemma 6 suggests if the weight growsslower than n 1 / 5 here is no limit to the number of codewordsthat can be found.Finally, we demonstrate that when the auto- and cross-correlation constraints are growing like nk for a constant k , weare guaranteed the existence of codes provided the codewordweight grows no faster than fi.Lemma 7: Let p , q , and r be constants 0 < p , q , r < 1.Then if p < 1 / 2

for any positive real cy, p, and y.Proof: See Appendix 111.Iv. OOCS AS CONSTANT-WEIGHTUNEQUAL RROR ROTECTIONODES

In this section we briefly describe the connection between( n ,w , A,, A) optical orthogonal codes and unequal errorprotection (UEP) code with two levels of protection.An unequal error protection code is an error control codewith a twist; the code is designed so that different digitsin a message have varying levels of reliability. This maybe convenient when the position of a digit in a messagedetermines its importance. The archetypical example is amessage containing a bank balance; if the balance is $1376.62it is much more important that the 1 be uncorrupted thanthat the 2 be error-free.An encoder for an ( n , k ) binary error control code isa mapping f : 0, l} + (0, l} . The message z E(0, l} s represented by the codeword f ( z ) E (0, l}.If min ( d ( f ( z ) , f (y)) : z, E (0, l}k, z y} 2 2t + 1 thenwe say the code is t-error correcting. (Here, d ( c 1 , c2 ) is theHamming distance between the n-tuples c1 and c2.)

Dejinition: Given an encoder f : 0, l} k -f (0, l} , heseparation vector associated with the encoder is an integerk-tuple s = [so, S I , . . ,S k - 1 1 defined by

s, = min { d ( f ( z ) , f (y)) : z, E (0, and z, # y t } .Note that the separation vector is associated with the en -coder rather than the code-i.e., the image of the encoder. Itis possible that a code may have multiple separation vectorsassociated with it4orresponding to different encoders for thesame code.Let f ( . ) be an encoder with separation vector s =

[so, 5 - 1 , . . . ,S k - 1 1 . Suppose a message k-tuple z s used toselect a codeword f (z) E C which is then transmitted overa noisy channel. Minimum-distance decoding will correctlyrecover the ith bit of the message provided no more thant , = [(s, - 1)/1] errors occur during transmission. A codewith an encoder whose separation vector has the property thatt , # t, for some i and j is called an unequal error protection(UEP) code.There is a substantial body of literature on UE P codes.(See [8]-[ 111 for references.) However, there has been noinvestigation of constant weight UEP codes.An (n , w, A A OOC with A > A can be used toconstruct a constant-weight UEP code. Suppose you have suchan OOC with cardinality M . Now consider the error controlcode consisting of all the n-tuples of the OOC and all theircyclic shifts. The resulting code has nM codewords of weightw. Furthermore, such a code consists of M clouds, wheretwo codewords belong to the same cloud if and only if theyare cyclic shifts of one another. The distance between anytwo codewords within the same cloud is at least 2(w - A,)the distance between any two codewords from two differentclouds is at least 2(w - A,).So , consider the following encoder. Take IC2 = [log, M Jmessage bits and use them to pick a cloud; then take k l =[log, nJ message bits and use them to pick an n-tuple fromwithin the chosen cloud. Any two messages that differ in the

first k2 bits will have codewords that differ in at least 2(w-A,)positions; any two messages that differ in the last k l bits willhave codewords that differ in at least 2(w - A,) positions.Therefore, we have described an encoder for an ( n . k l + kp)code with separation vectors = (2(w - A,), . ,2(w - A,), 2(w - A,), . . 2 ( w - A .)

\ + /kz ki

This observation means that our lower bound for OOCs maybe interpreted as an existence result for constant-weight UEPcodes with two levels of protection.Notation: Let M ( n , w, A,, A,) denote the lower bound on

@(n. , A A derived in Theorem 3-i.e., M ( n , w, A,A) = ( (z ) - A ) / B ,where A and B are given in Theorem3. Theorem 4: Let cy and /3 be positive, even integers. Thenthere exists a weight-w ( n , k 1 + k 2 ) error control code withseparation vector

3 = 1% Q , . , c y , ,p,PJ.,g:-2 ki



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A. Construction I70 -

60 -

50 --

40 -

30 -

20 -

10 -

0 ~ 1 ~ ~ ~ 1 1 1 0 20 40 60 80 100n (Blocklength)

Fig. 1 . Lower bounds on the number of message bits that can be protectedagainst two errors while [log, n] bits are protected against single errors,assuming codewords of weight L n / 2 ] .

wherekz = [log, M ( n ,w, - P / 2 ) ,w - Q /2 ) ) lkl = [log, n ] .

and

Fig. 1 uses this result to bound the number of informationbits that can be protected against two errors while simulta-neously protecting kl = [log, (n ) ] nformation bits againstsingle errors. This is done as a function of blocklength n ,assuming the weight of each such codeword is Ln/2].

V . CONSTRUCTINGN ( n ,W , 2, 1) OOCIn this section we present two techniques for constructing

( n ,w , 2, 1) optical orthogonal codes. Both are derived frompreviously published methods for constructing a boundedincomplete block design (BIBD). A BIBD is a structure equiv-alent to an (n ,w , 1, 1) OOC; the techniques outlined belowloosen the autocorrelation constraint to two, with a resultingdecrease in blocklength when compared to the associated( n ,w , 1, 1) code.

This method is a variation on the technique proposed byWilson [13] to construct ( n ,W , 1, 1) codes. We begin withthe case w = 5, and then generalize the technique.An ( n ,5, 2, 1) OOC: Let n be a prime number such that

n = 12t + 1 for integer t . Let Q be a primitive element ofthe field GF (n ) such that a4 = cy3t - 1 and ar = 2, where qand r are integers that are nonzero modulo-3 and are distinctmodulo-3.Then we can construct an ( n ,5, 2, 1) OO C C with car-dinality (CI = t as follows. The ith codeword z, ontainsa 1 in positions 0, a3, c ~ ~ ~ + ~ ~ ,~ ~ ~ + ~ ~ ,nd agt+ anda 0 everywhere else. This holds for i = 0, 1, -1. (Note: We say that the code consists of the blocks

To see that this construction yields an ( n , , 2 , 1) codelet Rz , denote the array consisting of all the relative delaysbetween pairs of 1s in 20. eeping in mind that alZt= 1 and@ = -1 , simple algebra reveals that (see bottom of page)Every component of Rz , is of the form Pa3jt whereP E (1, 2, d - l } a n d j E (0, 1. 2, 3}.Therefore,aslongasthe base-cY logarithms of 2 and Q~~- 1 are nonzero modulo-3and not equivalent to each other modulo-3

, 1 1: 2 = 0 , 1 , . . , - }.)[o, Q3t+3z a6t+3z a9t+32

Hence, no element of Rz a appears more than twice; thisimplies the autocorrelation constraint for zo s met. AndRz, = a3 . Rz0 so the autocorrelation constraint is met forall 2,.To check the cross-correlation constraint, we note thatM 2, is a set whose vectors are just the 1-tuples fromRz. Therefore, as long as the components of Rz, and thecomponents of RZt, form disjoint sets for i # i, we willhave proven the cross-correlation constraint is met. But (asmentioned above) RE, s obtained by multiplying R,, by a3;thus the components of Rz, are of the form P Q ~ ( J ~ + ) ;s longas 0 5 i 5 t - 1 he components form disjoint sets and so thecross-correlation constraint is met.

Example: Let n = 37 and t = 3. Choosecy = 2 as the primitive element of GF (3 7) and so23t - 1 = 30 = 214 while 2 = 2l-i.e., q = 14and T = 1. Then the code consists of the blocks{[0, 1, 6, 31, 361 [0, 8, 11, 26, 291, [0 ,10, 14, 2 3 , 27]}and so the three codewords are

2 0 = [1100001000000000000000000000000100001]2 1 = [1000000010010000000000000010010000000]2, = [1000000000100010000000010001000000000].

Note that, using Theorem 2, we know that Q(37, 5 , 2, 1) 5(36. 2 ) / ( 5 4) = 3.6 and so this construction is optimal.



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The above construction can be generalized to other valuesof w. The constructions are given below.An ( n ,w, 2 , 1 ) OCC fo r Even w : Let w = 2 m and choose

n to be a prime number such that n = w 2 t / 2 + 1 for aninteger t . Let cy be a primitive element of G F ( n ) such that{log, [akmt 11: 1 5 IC 5 m} are all distinct modulo-m andnonzero modulo-m. Then the code consisting of the blocksI :[ ( p i ; p ( i + t ) a m ( i + 2 t ) . . c y m ( i + ( 2 m - 1 ) t ). l i = 0 , l , . . . , t - l }

is an ( n ,w, 2 , 1) OOC.An (n .w, 2 , 1) OOCfor Od d w: Let w = 2m + I andchoose n to be a prime number such that n = (w 2- ) t / 2 + 1for an integer t. Let cy be a primitive element of GF(n) uchthat {log, - 1:1 5 IC 5 m ) are all distinct modulo-

m + 1 and nonzero modulo-m + 1. Then the code consistingof the blocks

B. Construction 2We now demonstrate another technique for constructing an

(n ,w , 2, 1) optical orthogonal code. This technique is basedon the approach of Hanani [14]. As before, we first provethe construction works for w = 5 and then present (withoutproof) the generalization.An (n ,5 . 2, 1 ) 0012: et n be a prime number such that

n = 1 2 t + l for an integer t . Let a be a primitive element of thefield G F ( n) uch that, for some integer y E {1, 2 , . . . 6 t -1)all of the following are nonzero and distinct modulo-6:log,[cyY - 11log, [a6t- 11log, [ a 6 t - 11log, [ c y Y ( Q 6 t - ) ] .

Then we can construct an (n ,5, 2, 1)OOC C with cardinality IC1 = t from theI :locks {[o, a 6 2 , cyy+621 (r6t+6z. C y 6 t+ 6 z + y

L = 0. l , . . ' , t - }. )A s before, let Rz, be the relative delay array for xo.Then

RZ, is (see bottom of page)Every component of Rzo is of the form pa6Jt where[j E {I, a y , a y - 1, c y G t - a y , - 1, aY(aGt- 1)) andj E (0, ) . Therefore, as long as the base-a logarithms ofa y , a y - 1, - a y , cy6t - 1, and ay(ast - 1 ) are notequivalent to each other modulo-6 and are not equivalent to

zero modulo-6

Therefore, no element of Rz, is repeated more than once;this implies the autocorrelation constraint for zo is met.Furthermore, the matrix RZ, is obtained by multiplying Rx0by a6',so the autocorrelation constraint is met for all z,.The crosscorrelation property is proven as in the construc-tion in Section V -A .This construction is generalized below.An ( n ,w, 2 , 1) OOC fo r w = 4m : Let n = w 2 t / 2 + 1 bea prime number, where w = 4m. Furthermore, assume that ais a primitive element of G F ( n) uch that all of the followinghold for some integer y, 1 5 y 5 4 m t - 1:

c y k 4 m t + y - 1= a z k ,a k l m t - a y = c y ~ k , for IC = 0 , 1 , . . m - 1for IC = 1, 2 , . . . , m

0 @mt - 1 = a r k ,c y ~ ( a ~ ~ ~ ~1 ) = a s k ,

Here, the integers i o , i l , . . , - 1 , jl, . , ,, T I , . . ,Tfor IC = 1, 2 , . . . , mfor IC = 1, 2 . . . . , m .

and s l , . . ,s are all distinct modulo-4m. Then the blocks{ [ a 4 m z , a y + 4 m z a 4 m t + 4 m z a 4 m t +y + 4 m a . .

1 , , >a 4 m ( 2 m - l ) t + 4 m z a 4 m ( 2 m - l ) t+ y + 4 m z ] : i = O , l , . . . , t - l }are the codewords of an (n ,w, 2 , 1) OOC.An ( n ,w , 2, 1) OCC for w = 4 m + 1: Let n = (w 2 -l ) t / 2 + 1 be a prime number, where w = 4 m + l . Furthermore,assume that a is a primitive element of G F ( n ) uch that all ofthe following hold for some integer y , 1 5 y 5 (4m + 2 ) t - 1:

0 c y k ( 4 , + 2 ) t + ~ - 1= cya k ,0 c y k (4 m + 2 ) t - ay + c y j k ,0 a k ( 4 m + 2 ) t - 1 = a r k ,0 a y ( c y k ( 4 m f 2 ) t - 1 ) = a s k ,Here, the integers y , i o , 2 1 , . . . , m - 1 , j i , . . , ,, T I , . . , m ,

for = 0, 1 , . ,m - 1for IC = 1, 2 , . . . ,mfor IC = 1, 2 , . . ,mfor IC = 1, 2 , . . ,m.and SI, . . .sm are all distinct modulo-4m + 2 and nonzeromodulo-4m + 2. Then the blocks{ [o , x(4m+2)z x~ +(4 m+ 2)z x(4m+2)t+(4m+2)z> , 1

x ( 4 m + 2 ) t + y + ( 4 m + 2 ) z , . .1 ,

I:4 m + 2 ) ( 2 m - 1 ) t + ( 4 m + 2 ) z l , (4m+2)(2m-l) tfy+(4m+2)zzi = o , l , . . . , t - l )

are the codewords of an ( n ,w, 2 , 1 ) OOC.Example: Let n = 41, w = 4, and t = 5 . Choosea = 6 as the primitive element of G F ( 4 1 ) and choose

y = 3 and so 6 Y - 1 = 1 0 = 68, 6" - 6 y = 29 = 67 ,620- 1= 39 = 6 6 , and cy(6'O - 1)= 1 9 = 6g-i.e., i o = 8,j l = 7, r1 = 6, and s1 = 9. Furthermore, io, ~ ,I , S Iare distinct modulo-4. Then the code consists of the blocks



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YANG AND FUJA: OPTICAL ORTHOGONAL CODES

~

103

TABLE ITH ECARL HNAL I T YF ( n , LL 2 , 1) OOCs CONSTRUCTEDSI NG HE TECHNIQUEF SECTION-A

TABLE I1TH ECARDI NAL I T YF (71, LL 2 , 1 ) OOCs CONST RUCT E DSING HE T E CHNI QUEF SECTION-B

{[ l , 1, 30, 401, [12, 16 , 25, 291, [ l o , 13 , 28, 311, [3, 4, 37,381, [7, 18, 23, 34]}-i.e., the codewords are2 0 = [01000000000100000000000000000010000000001]5 = [000000000000 0001000000001000100000000000]5 2 = [00000000001001000000000000001001000000000]5 3 = [OOO1 00000000000000000000000000000000 1 00124 = [00000001000000000010000100000000001000000].Note that, from Theorem 2, a(41,4, 2, 1) 5 2 .40/12 =6.66, so we cannot say for sure if this code is optimal; theremay be a (41, 4, 2 , 1) code with six codewords.Tables I and I1 show the parameters of some ( n ,w, 2, 1)OOCs that can be contructed using the approaches outlinedabove. Also included is the upper bound on @ ( n , , 2, 1)derived in Theorem 2. Theorem 2 tells us it is impossibleto construct an ( n ,w, 2 , 1) OOC with more than 2 ( n -l )/ (w(w - 1)) codewords; the methods above tell us howto construct ( n ,w, 2, 1) OOCs with 2(n- )/ w2 codewords

(for even w) as well as ones with 2(n- 1)/(w2- 1)codewords(for odd w).These constructions also illustrate the point made in Section11-B that, for some blocklengths, codes with A # A maybe preferable to codes with equal constraints. Comparingan ( n ,w, 1, 1) code with an ( n ,w + 1, 2 , 1) code, recallfrom Section 11-B that the performance figures of merit forthe ( n ,w + l , 2, 1) code dominate those of the ( n ,w, 1, 1)code. Yet we have just shown it is possible to construct an(71 , w +l , 2 , 1) code with either Z( n - l ) / ( ~ + l ) ~odewords(for even w + 1) or 2(n- )/w(w +2) codewords (for oddw+ l) ; but Theorem 1 tells us it is impossible to construct an( n ,w, 1, 1) code with more than (n -l )/ (w (w l) ) codewords.Therefore, for w 2 6 the ( n ,w + 1, 2, 1) codes offer betterperformance and more codewords than any ( n ,w, 1, 1) code.As a simple example, from Table I1 we see that it is possibleto construct a (1801, 9, 2, 1) OO C with 45 codewords. Butto construct an (1801, 8, 1, 1) code the Johnson bound tellsus it is impossible to have more than 32 codewords.



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VI. SUMMARYIn this paper we derived new bounds on the number of usersthat can be supported on an optical network employing code-division multiple access with binary signature sequences; inaddition we presented a number of new methods for designingcodes with good auto- and cross-correlation properties. Unlikeprevious work in this area, we considered the possibility

that the auto- and cross-correlation constraints might not beidentical; indeed, the bounds and the constructions suggestthat it may sometimes be preferable to use such asymmetricOOCs. Among the constructions presented, we note thatthe ( n ,w, 2 , 1) codes are near-optimal; their cardinaltiy is2 ( n - 1)/w2 and we have demonstrated that it is impossibleto get more than 2( n- 1)/(w2 - w) odewords.We also noted the relationship between OOCs with unequalconstraints and constant-weight unequal error protection codeswith two levels of protection; we noted that the lower boundwe derived for OOCs could be interpreted as a lower boundon the achievability of such UEP codes.

APPENDIXLemma 4: Let z E C,where C is an ( n ,w, X + m, A)optical orthogonal code. (Assume m 2 0 is an integer.) Then

X + m ., A I -Proof: Let

Define a function f : Z -+ 2 byr in 2 ,

where tZ = [ t o , t l , . . . , t , - l ] .is exactly the image of f ( . ) ; o prove the lemmait is sufficient to show that f ( .) maps at most m + X elementsof Z o the same element of Mz A; then we will have shownI M Z , ~ I2 lxl/(m + A) = w ( f l ) / ( ~+ m).Now define a function f : Z + 2 uch that g ( i ) is the sumof the components of f(i)-i.e., if f( i) [ao ,a l , . .,ax-I]then g ( i ) = a0 + a1 + . . .+ ax-1. Obviously, if f(i) f($)then g ( i ) = g ( ) so to prove the lemma it is sufficient toshow that g ( . ) maps at most m + X elements of Z to the sameinteger. For each j E Z the integer g ( i ) is a component of Rz,and we know no component of RZ can be repeated more thanX + m times by Lemma 1. Therefore, the function g ( . ) is (atmost) (m + A)-to-one and the Lemma holds. Q.E.D.

So Mz,

APPENDIX 1We now prove that the number A in Theorem 4 is an upperbound on the number of binary n-tuples of weight w withautocorrelation exceeding A,.Associate the w-set S = {sl , s2, . . ,s,} with the biaryn-tuple containing ones in positions s1, s 2 , . . ,s , and zeroeseverywhere else. We wish to (over) count the number of w-

sets associated with n-tuples that violate the autocorrelationconstraint.Fix S to be a positive integer. Then a chain{ i o , i l , . . . i, } is a set of integers modulo-n wherei, = i,-l + 6 for 1 5 j 5 z; the length of this chainis 1 + z. By convention, a cycle (i.e., io = i, + 6) isconsidered a chain of length z + 1 with an arbitrary startingpoint. A maximal chain is one not contained in another chain.Now suppose the w-set S = {SI, s 2 , . . . ,s,} can bepartitioned into c maximal chains whose lengths are 1+21, 1+5 2 , . . . ,1 + z,. Then clearly

Cw = c +cza.i = l

Realize that the w-set S is specified exactly by:1) The value S upon which the partitioning chains are2 ) The number c of maximal chains into which it can be3) The lengths of the maximal chains-i.e., X I , 5 2 , . . . , x C .4) The chain heads-i.e., the starting point of each chain.The autocorrelation of the n-tuple associated with S after Scyclic shifts is w- + N , where N is the number of chains thatare cycles. This is because a cycle of length 2,+1 adds 2 , +1to the autocorrelation, whereas a noncyclic chain of length

2 , + 1 adds only z,. Our approach, therefore, will be to countthe number of w-sets with the property that w - c + N > A,;in doing so we will let S vary from 1 to Ln/2]. For 6 > n/2we shall have already counted the associated w-sets with

Once n ,w, and S are fixed, the only way a chain can bea cycle is if S l kn for some integer k . Furthermore, if Slknbut 6 n then there is some other value 6 such that 6/16 andSln and the cycle associated with S is identical to a cycleassociated with 6. Therefore, in looking for cycles we needonly look at values of S hat divide n; when S n we will notfind any cycles that have not already been accounted for.

So how many cycles can there be? Each cycle uses upn/S of the w ones; therefore, there can be anywhere fromzero to Lw6/nJ cycles-i.e., 0 5 N 5 LwS/nJ.So, suppose we have fixed n, w, 6, nd N . We are nowgoing to specify the chains. How many ways are there to pickc nonnegative integers 5 1 , 5 2 , . . z, such that z1 + 5 2 +. . + 2, = w - and w - c + N > A,. Note that c must beat least [wS/nl; this is because the smallest number of chainsis caused by making each chain as large as possible, and thelargest chain is a cycle. Thus c is smallest when there areLwS/n] cycles and (possibly) one leftover noncyclic chainfor a total of [w6/nl chains.

based.partitioned.

6/ = n - 6.)



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So now suppose we have fixed n, w, 6, , and c. Then thereare (G) ays to pick the chains that are cycles. Furthermore,of the xis o be equal to (n /S ) 1; thus it remains to pick theremaining c- N xis o add up to w - - N ( n/S)- 1). For

thatA < fnp(2Aa+3)-Xa a2Xa+32(A, + l)!,nce we have picked those N cycles we have fixed exactly N - W!

andc > N there are ( c - N - l ways to pick these c - Nxis.Note that the term L ( N -wS/n) is included to count the casew-Nnh- l )where there are c = N = wS/n chains and they are all cycles.For the case c = N we must have c = N = w S / n , since allof the 1s are used up in chains, and in this case the question

Since p < A,/(2A, +3) , A is a low-order term compared withnw/w!. ombining the numerator and denominator, we haveof picking the remaining xis becomes vacuous.The last terms in the first sum are used to count the numberof ways to pick the heads of the chains. We know that anycycle must begin in one of the first S positions; we knowfurthermore that the c- N noncyclic chains may begin in anyof the n - Nn/S positions not taken up by cycles.The second sum counts the number of w-sets violating theautocorrelation constraint when S 1 n and so there can be nocycles. There are ( ways to pick these cz,s to add upto w - c and (I)ways to pick c chain heads. Therefore, Arepresents an upper bound on the number of binary n-tuples

+ (low order terms).+ (low order terms).

Since p < A,/(2A, + 3) we obtain the desired result. Q.E.D.Lemma 7: Let p , q , and r be constants, 0 < p , q , r < 1.Then if p < l / 2that violate an autocorrelation constraint of A Q.E.D. lim @(n, anpj , ( Pn q J , 7nJ)= 00n+m

APPENDIX11 for any positive real a , p, and y.Pro08 Define A,(n) = [flnqj and A,(n) = Lyz J . ThenLemma 6: Let A and A be positive integers, and let p be a

lim @ ( n , anP],A A = ccn-03Noting this, the result can be proved using an approachessentially identical to that used in proving Lemma 6. If p


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I06 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 1, JANUARY 1995

[ I O ] M. C . Lin and S . Lin, Cyclic uequal error protection codes constructedfrom cyclic codes of composite length, IEEE Trans. Inform. Theory,vol. 34, pp. 867-871, July 1988.[ I 1 1 R. H. Morelos-Zaragoza, Multi-level error correcting codes, Ph.D.dissertation, Univ. of Hawaii, May 1992.[121 L. A. Bassalygo, V. A . Zinovev, V. V . Zyablov, M. S. Pisker, and G .S. Poltyrev, Bounds for codes with uequal protection of two sets of

messages, Probl. Pered. Inform., vol. 15, pp. 40 4 9, July-Sept . 1979.[131 R. M. Wilson, Cyclotomy and difference families in elementary abeliangroups, J . Number Theory, vol. 4, pp. 1 74 7, 1972.[141 H. Hanani, The existence and constructions of balanced incompleteblock designs, Ann. Marh. Sfarisr., vol. 32, pp. 361-386, 1961.[15] R. C. Bose, On the construction of balanced incomplete block design,Ann. Eugenics, vol. 9, pp. 353-399, 1939.

Optical Orthogonal Codes With Unequal Auto-correlation and Cross-correlation Constraints

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