A Survey on Optical Orthogonal Codes

Transcription

1 A urvey on Optil Orthogonl Codes Mohd M. Ale-Krldni Optil orthogonl odes (OOC) defined by lehi [1] nd Chung, lehi, nd Wei [] re fily of (0,1) seuenes with desired utoorreltion nd ross-orreltion properties providing synhronous ulti-ess ounitions with esy synhroniztion nd good perforne in OCDMA ounition networks [1,3]. In this setion we review few of the ost iportnt lgoriths in generting OOCs. I. Mthetil Forultion An optil orthogonl ode n, w, λ, λ ) is fily C of (0,1) seuenes of length n with ( onstnt Hing weight w stisfying the following two properties: 1. Autoorreltion property. For ny odeword x = ( x 0, x1,..., x n 1) C, the ineulity n 1 xx i i τ λ holds for ny integer τ / 0(od n) i = 0 1, nd. Cross-orreltion property. For ny two distint odewords xy, C n 1 x iy i τ λ holds for ny integer τ, i = 0, the ineulity where the nottion denotes the odulo- n ddition []. When λ = λ = λ, we denote the OOC by ( nw,, λ ) for sipliity. The nuber of odewords is lled the size of the optil orthogonl ode. Fro prtil point of view, ode with lrge size is reuired [3]. To find the best possible odes, we need to deterine n upper bound on the size of n OOC with the given preters. Let Φ n, w, λ, λ ) be the lrgest possible size of n n, w, λ, λ ) OOC. An ( ( OOC hieving this xiu size is sid to be optil. It is esily shown tht if ww ( 1) > λ ( n 1) then Φ ( nwλ,,, λ ) = 0 nd if w > λ n then Φ( nwλ,,, λ ) 1 [4]. Bsed on the Johnson bound for onstnt-weight error orreting odes [7], we hve the following bound []:

2 where the nottion 1 n 1 n n λ + 1 n λ Φ ( n, w, λ) w w 1 w w λ + 1 w λ denotes the integer floor funtion. Also, it is ler fro definition tht Φ( nw,, λ, λ ) Φ ( nw,, λ) where λ = x { λ, λ }. As n exple, the following two seuenes re the odewords of (13,3,1) OOC []. (1) C = { , } () This ode is optil sine Φ(13, 3,1) 3 =. Another useful depition ethod for OOCs is the set-theoretil representtion { ; 1} X = k Z x = for eh odeword = ( x0, x1,..., xn 1) n k x, where Z = { 0,1,..., n 1} denotes the odulo- n integers []. For exple, the (40,4,1) OOC n be represented s: {{ } { } { }} (40,4,1) OOC = 0,1,8,37, 0,,18,5, 0,5,11,19 (od 40) Then the orreltion properties for every odeword X nd Y n be reforulted s follows: Autoorreltion property: ( X) ( b X) λ b (od n) / Cross-orreltion property: ( ) ( ) X b Y λ where X is eul to { x : x X } nd X denotes the rdinlity of the set X. n (3) II. Constrution Methods There re severl ethods for onstruting OOCs tht n be tegorized into two ses [], one is diret ethods whih use the thetil strutures suh s projetive geoetry [,16], finite field theory [4,1] nd design theory [4,8,9,17], nother is serh ethods whih use the oputer lgoriths suh s greedy nd elerted greedy lgoriths [] nd outer-produt trix lgorith [18].

3 The first thetil design ethod of OOCs, presented in the originl pper on OOC [], is bsed on finite projetive geoetry. There is n ( nw,,1) OOC orresponding to projetive d 1 1 geoetry PG( d, ) where d is positive integer nd is prie power suh tht + n = 1 nd w= + 1. Eh odeword orresponds to line in PG( d, ) where eh line is obtined fro plne rossing the origin in ( d + 1) -diensionl vetor spe on Glois field GF( ). It ws shown tht, [], the nuber of odewords obtined fro this ethod is eul to d 1 1 for even d nd d 1 when d is odd whih hieve the Johnson bound in eh two ses. Therefore, this ethod gives n optil ( nw,,1) optil orthogonl ode. For exple, the settheoretil representtion of optil (341,5,1) OOC obtined fro PG (4, ) with 17 odewords is given in Tble I []. Tble I- Codewords of (341,5,1) OOC

4 Due to the Johnson bound, OOCs with λ = 1 hve fewer nuber of odewords nd therefore few nuber of users n be oodted in the orresponding OCDMA networks. Hene, OOCs with λ, λ > 1 whih hve ore odewords nd re soeties lled generlized OOC hve been exined in OCDMA systes. urprisingly, it ws shown in [5], s n exple, tht for 50 users, the (1000,1,) OOC hs better perforne thn (1000, 5,1) OOC. This point hs been deeply nlyzed nd verified in [6] nd it ws shown tht OOCs with λ =,3 ould hve better perforne thn odes with λ = 1. Conseuently, the onstrution ethods of generlized OOCs hve found speil iportne. The first onstrution ethod of optil orthogonl odes with λ = is due to Chung nd Kur [4] whih uses finite field theory to design n optil ( p 1, p 1,) + OOC with p odewords where p is prie nuber nd is positive integer. Let α be priitive eleent of set-theoretil representtion of odewords is s follows, GF p ( ) nd p 1 =, then the β α { α β } p i = log ( x) ; ( x 1) =, i = 1,,..., p i Tble II ontins the odewords of (63,9,) OOC obtined for p = nd = 3 [4]. (4) Tble II- Codewords of (63,9,) OOC 1 { 1,5,8,18, 8,31,35, 40,59 } {,7,10,16,17,36,55,56,6 } 3 { 3,11,4,5,7,9,30,43,51 } 4 { 4,9,14, 0,3,34, 47, 49,61 } 5 { 6,,3,39,48,50,54,58,60 } 6 { 1,15,33,37,44,45,46,53,57 } There is strong reltionship between OOCs nd onstnt-weight Cylilly Perutble Codes or CPCs. In the other words, n ( nwλ,, ) OOC is euivlent to the ( n,w λ, w) CPC whih indites n error-orreting ode of length n, weight w, nd iniu Hing distne w λ. Every yli shift of CPC odeword is lso odeword [10]. o, using 4

5 onstrution ethods of CPCs we n design new OOCs. In [10] severl suh ethods espeilly for λ = 1 hve been exined. OOCs with uneul uto- nd ross-orreltion onstrints hve been investigted in detil by Yng nd Fuj [8]. A new upper bound useful for se λ > λ presented in [8] is s follows: ( λ + )( n 1)( n )...( n λ) Φ ( nw,, λ+, λ) (5) ww ( 1)( w )...( w λ) For exple, fro the Johnson bound for (41, 4,,1) OOC we hve Φ(41, 4,,1) Φ(41, 4, ) 63 while fro the bove bound we hve Φ(41,4,,1) 6, whih indites tht it is ipossible to hve ore thn 6 odewords for (41, 4,,1) OOC; so, for λ > λ it is tighter thn Johnson Bound. Another useful ethod for OOC design is presented in [1], whih uses two thetil strutures, nely, Perfet Differene et (PD) nd finite Mobios Geoetry (MG). A k-subset = { d d } of Z { 0,1,..., n 1} D 1,,..., d k n = is lled n ( n, k, λ ) PD whenever for every / 0 (od n ) there re extly λ ordered pirs ( d, d ), i j suh tht d d (od n). A finite i Mobios geoetry MG (, r) with prie power nd positive integer r is n extended Glois field GF( r ) { } 5 j with ll irles on it. Bsed on one-to-one orrespondene between r r r ( + + 1, + 1,1) -PD nd MG (, r), n optil orthogonl ode with preters ( r r + + 1, + 1,1,) nd size r 1 r 1 1 i n be obtined. As n exple, if = nd r =, we hve ( 1, 3,1, ) OOC with 0 odewords [1]. The dvntge of this ethod is in its bility to generte lrge nuber of odewords, but the low weight reuireent is jor drwbk for this struture. There re soe reursive onstrutions for OOCs [,11]. One of the best reursive ethods for ( nwλ,, ) OOC design ws obtined by Chu nd Golob [13]. This ethod uses r -siple tries over yli group for reursive onstrution of OOCs whih indites tht if there exists n ( nwλ,, ) OOC with T odewords, then there exists n ( n, w, λ ) OOC with T λ odewords whenever the prie ftors of re not less thn w. As n exple, for = 11 nd j

6 (63,9,) OOC with 6 odewords, this ethod onstruts (693,9,) OOC with 6 11 = 76 odewords [13]. One of the dvntges of this ethod is tht if the ( nwλ,, ) OOC is optil then the onstruted ( n, w, λ ) OOC is t lest syptotilly optil [13]. Cylotoi lsses nd nubers with respet to the finite field GF( ) re the thetil strutures tht n be used to onstrut OOCs. Using this ethod, Ding nd Xing presented severl lsses of ( 1, w,) OOCs [14]. Also, five lsses of ( 1, w,) OOCs where is power of odd prie hve been derived using ylotoy [15]. By using finite projetive geoetry, we n lso design OOCs with λ > 1. In [16] bsed on onis on finite projetive plnes in the projetive geoetry PG(3, ) n syptotilly optil 3 ( 1, 1,) OOC with 3 + odewords hve been obtined. For exple, if = 3, the (40,4,) OOC with 1 odewords is obtined [16]. Blned Inoplete Blok Design (BIBD) is one of the ost beutiful strutures in disrete thetis tht hve lose reltion with OOCs. In other words, every ( nwλ,, ) OOC is euivlent to ( λ + 1) ( nw,,1) stritly yli prtil design [11,17]. ine there re severl ethods for onstruting BIBD suh s Wilson nd Hnni, we n use the to design OOCs. Wilson s ethod is pplied for designing of (, nw,1) OOC [4] nd ( nw,,,1) OOC [8]. Both ses re tegorized to even w nd odd w. In this ethod n is prie nuber obtined fro weight w nd the size of the ode. For exple, the following optil (37,5,,1) OOC with 3 odewords hve been obtined fro Wilson s ethod for odd w. {{ } { } { } } (37,5,,1) OOC = 0,1,6,31,36, 0,8,11,6,9, 0,10,14,3,7 ( od 37) Hnni s ethod lso like Wilson s is for designing of (, nw,1) OOC [9] nd ( nw,,,1) OOC [8]. This ethod for onstruting (, nw,1) OOC is tegorized into two ses, nely, w (od4) nd w 3(od4). But for the onstrution of ( nw,,,1) OOC it is divided in two ses w 0(od4) nd w 1(od4). In ll these ses the ode length n is prie nuber obtined fro ode weight nd the nuber of odewords. The (41, 4,,1) OOC with following 5 odewords re obtined fro Hnni s ethod [8]. (6) 6

7 { } { } { } { 3,4,37,38 },{ 7,18,3,34 } 1,11,30, 40, 1,16, 5, 9, 10,13, 8,31, (41, 4,,1) OOC = (od 41) Although it sees tht Hnni s ethod is speil se of Wilson s onstrution, there re soe ode lengths tht Hnni s onstrution yields odes while Wilson s ethod does not [9]. One of the best-known ses of BIBD is teiner Qudruple yste (Q). Bsed on Q, Chu nd Colbourn [17] suggested n lgorith for optil (,4,) n OOC with n 44. As n exple, optil (10,4,) OOC with 3 odewords obtined fro this ethod is s follows [17], (7) (10,4,) OOC= {{ 0,,4,7 },{ 0,1,3,4 },{ 0,1,,6 } } (od10) (8) III. Referenes [1]. J.A. lehi, Code division ultiple-ess tehniues in optil fiber networks-prt I: Fundentl priniples, IEEE Trnstions on Counitions, vol. 37, no. 8, pp , August []. F.R.K. Chung, J.A. lehi, nd V.K. Wei, Optil orthogonl odes: Design, nlysis, nd pplitions, IEEE Trnstions on Infortion Theory, vol.35, no. 3, pp , My [3]. J.A. lehi nd C.A. Brkett, Code division ultiple-ess tehniues in optil fiber networks-prt II: ystes perforne nlysis, IEEE Trnstions on Counitions, vol. 37, no.8, pp , August [4]. H. Chung nd P. Vijy Kur, Optil orthogonl odes-new bounds nd n optil onstrution, IEEE Trnstions on Infortion Theory, vol.36, no. 4, pp , July [5]. M. Azizoglo, J.A. lehi, nd Y. Li, Optil CDMA vi teporl odes, IEEE Trnstions on Counitions, vol. 40, no.7, pp , July 199. [6].. Mshhdi nd J.A. lehi, Code division ultiple-ess tehniues in optil fiber networks-prt III: Optil AND gte reeiver struture with generlized optil orthogonl odes, IEEE Trnstions on Counitions, vol. 45, no. 5, pp , August 006. [7]..M. Johnson, A new upper bound for error-orreting odes, IRE Trnstions on Infortion Theory, vol. IT-8, pp , 196. [8]. G.C. Yng nd T.E. Fuj, Optil orthogonl odes with uneul uto- nd rossorreltion onstrints, IEEE Trnstions on Infortion Theory, vol. 41, no. 1, pp , Deeber

8 [9]. G. C. Yng, oe new filies of optil orthogonl odes for ode-division ultipleess fiber-opti networks, IEE Proeeding on Counitions, vol. 14, no. 6, Deeber [10].. Bitn nd T. Etzion, Construtions for optil onstnt weight ylilly perutble odes nd differene filies, IEEE Trnstions on Infortion Theory, vol. 41, no. 1, pp , My [11]. R. Fuji-Hr nd Y. Mio, Optil orthogonl odes: Their bounds nd new optil onstrutions, IEEE Trnstions on Infortion Theory, vol. 46, no. 7, pp , Noveber 000. [1]. C.. Weng nd J. Wu, Optil orthogonl odes with nonidel ross-orreltion, Journl of Lightwve Tehnology, vol. 19, no. 1, pp , Deeber 001. [13]. W. Chu,. W. Golob, A new reursive onstrution for optil orthogonl odes, IEEE Trnstions on Infortion Theory, vol. 49, no. 11, pp , Noveber 003. [14]. C. Ding nd C. Xing, everl lsses of ( 1, w,) optil orthogonl odes, Disrete Applied Mthetis 18 (003) [15]. C. Ding nd C. Xing, Cylotoi optil orthogonl odes of oposite lengths, IEEE Trnstions on Infortion Theory, vol.5, no., pp , Februry 004. [16]. N. Miyoto, H. Mizuno, nd. hinohr, Optil orthogonl odes obtined fro onis on finite projetive plnes, Finite Fields nd Their Applitions 10 (004) [17]. W. Chu nd C.J. Colbourn, Optil (, 4, ) Mthetis 79 (004) n OOC of sll orders, Disrete [18]. H. Chrhi nd J.A. lehi, Outer-produt trix representtion of optil orthogonl odes, IEEE Trnstions on Counitions, vol. 54, no. 6, pp , June