Pattern Distributions of Legendre Sequences

Transcription

1 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY [9] J. E. Savage, Some imle elf-ychoizig digial daa camble, Bell Sy. Tech. J., vol., o.,. 9 87, Feb [10] A. Paouli, Pobabiliy, Radom Vaiable, ad Sochaic Pocee, d ed. ew Yok McGaw-Hill, 198. [11] J. G. Kemey ad J. L. Sell, Fiie Makov Chai. ew Yok Va oad, [1] W. R. Bee, Saiic of egeeaive digial amiio, Bell Sy. Tech. J., vol. 37, , ov [13] G. L. Caiolao ad G. P. Toca, Seca of block coded digial igal, IEEE Ta. Commu., vol. COM-, , Oc liea a of he Legede equece ake o oe of f 01 ; +1 ; 0 1; g, deedig o he value of mod 8. I hi coeodece, we develo lowe ad ue boud o he umbe of ae diibued i a cycle of a Legede equece, which i much ueio o kow boud. Ou eul how ha Legede equece have a ideal diibuio of ae of mall legh. We alo eablih he elaiohi bewee he weigh diibuio of quadaic eidue (Q.R.) code ad he ae diibuio of Legede equece. We ove ha evey biay eiodic equece wih kow ae diibuio give a cyclic code wih kow weigh diibuio; by kow ae diibuio we mea ha he umbe of all ae ee i a cycle of he equece ae kow. Pae Diibuio of Legede Sequece Cuheg Dig Abac Legede equece have a umbe of ieeig adome oeie ad ae cloely elaed wih quadaic eidue code. I hi coeodece we give lowe ad ue boud o he umbe of ae diibued i a cycle of he Legede equece ad eablih he elaiohi bewee he weigh diibuio of quadaic eidue code ad he ae diibuio of Legede equece. Ou eul how ha Legede equece have a ideal diibuio of ae of legh, whe i o lage comaed wih log, whee i he ime ued o defie he equece. Idex Tem Liea code, ae diibuio, equece, weigh diibuio. I. ITRODUCTIO Peudoadom equece have wide alicaio i agig yem, global oiioig yem, code-diviio mulile-acce yem, ada yem, ead-ecum commuicaio yem [6], [10], ad eam cihe []. Oe adome aec of equece i he ae diibuio. Oe of he hee oulae fo he adome of equece made by Golomb i abou he diibuio of ome ecial ae []. Le be a ime. The Legede equece l 1 wih eec o he ime i defied by l i 1+ i ; if i 6 0mod 0; ohewie fo each i 0, whee ( i ) i he Legede ymbol. Hee ad heeafe we ae dealig wih he f0; 1g veio of Legede equece. Legede equece have oimal balace bewee 1 ad 0. I i well kow ha he eiodic auocoelaio fucio of he Legede equece i wo-valued if 3 (mod ) ad heevalued if 1 (mod ). They have hu oimal auocoelaio oey. Dig, Helleeh, ad Sha [3] oved ha he Mauci eceived Febuay 17, 1997; evied Jue 8, The auho i wih he Deame of Ifomaio Syem ad Comue Sciece, he aioal Uiveiy of Sigaoe, Sigaoe ( digc@ic.u.edu.g). Publihe Iem Ideifie S (98)0383-X. II. PATTER DISTRIBUTIOS Le w 1 be a biay equece of eiod. Hee i o eceaily he lea eiod. Le Z f0; 1; 111; 0 1g be he eidue ig wih eec o iege addiio ad mulilicaio modulo. Defie he e C i fj Z w j ig fo each i 0; 1. Le i 1;i ; 111;i 01 f0; 1g ad 0; 1; 111; 01 be aiwie diic eleme of Z. Defie D i 111i ( 0; 111; 01) d i 111i ( 0 ; 111; 01 ) 01 k0 01 k0 (C i + k ) (C i + k ) The fo a fixed ad a e of fixed 0; 111; 01 he e fd i 111 i ( 0; 111; 01) i 0; 111;i 01 f0; 1gg (1) fom a aiio of Z. The above aamee d i 111 i ( 0 ; 111; 01 ) meaue he umbe of ae diibued i a cycle of he equece. A ae i defied o be a ig i i i 01 whee i 0; 111;i 01 ae fixed bi, hoe 3 ae do-o-cae bi ha could be eihe 0 ad 1, ad he diace amog i 0 ;i 1 ; 111;i 01 ae fixed. The diibuio of ecial ae of maximum-legh equece, i.e., block ad ga, wa udied by Golomb []. I wha follow we develo lowe ad ue boud o he aamee d i 111 i ( 0; 111; 01) fo Legede equece. By he defiiio of Legede equece, C 1 i he e of quadaic eidue modulo, ad C 0 Z C 1. Thu ae diibuio of Legede equece ae acually he diibuio of ae of quadaic eidue ad oeidue. Fo he diibuio of ae of legh wo i Legede equece we have he followig exac eul, which mea ha Legede equece have he be oible diibuio of ae of legh wo /98$ IEEE

2 169 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY 1998 Pooiio 1 If 3(mod), he d ij ( 0 ; 1 ) whee If 1 (mod ), he d 11( 0; 1) d 10 ( 0 ; 1 ) d 01( 0; 1) d 00 ( 0 ; 1 ) ( 0 3); fo (i; j) (1; 1) ( +1); fo (i; j) 6 (1; 1); 0 5 ; fo 0 1 eleme of Z 0 1 ; fo he ohe half eleme +3 ; fo 0 1 eleme of Z 0 1 ; fo he ohe half eleme +3 ; fo 0 1 eleme of Z 0 1 ; fo he ohe half eleme +3 ; fo 0 1 eleme of Z 0 1 ; fo he ohe half eleme whee Poof Le D 0 ad D 1 deoe he e of quadaic eidue ad oeidue modulo, eecively. Hee 0 i eihe a quadaic eidue o a quadaic oeidue. The cycloomic umbe of ode wo ae defied by (i; j) j(d i +1) \ D j j Thu hee ae a mo fou diffee cycloomic umbe of ode wo. By defiiio C 1 D 0 ad C 0 D 1 [f0g. oe ha D 0 i a mulilicaive gou. We have d 11 ( 0 ; 1 )j(d ) \ (D )j j(d 0 + ) \ D 0 j j(d j +1) \ D j j (j; j) whee , 01 D j fo ome j. Similaly, d 01 ( 0 ; 1 )j(d 1 [f0g + 0 ) \ (D )j j(d 1 [f0g + ) \ D 0 j j(d (j+1) mod + 1) \ D j j + jfg \D 0 j ((1 + j)mod;j)+jfg \D 0 j whee ad 01 D j fo ome j. Wih imila agume, we have ad d 10( 0; 1)(j; (1 + j) mod ) + jf0g \(D 0 + )j d 00( 0; 1)((j +1)mod; (j +1)mod) + jf0g \(D 1 + )j + jfg \D 1 j whee ad 01 D j fo ome j. If 1 (mod ), he cycloomic umbe of ode wo ae give by [11] (0; 0) 0 5 If 3 (mod ), hey ae give by (0; 0) (1; 0) (1; 1) 0 3 (0; 1) (1; 0) (1; 1) 0 1 (0; 1) +1 oe ha 01 D 0 if ad oly if 1 (mod ) ad D i if ad oly if 01 D i. The cocluio of hi ooiio he follow fom he cycloomic umbe of ode ad he above fou fomulae. I i oed ha he above ooiio ca alo be oved wih he followig Jacobhal fomula [1] 01 x0 01 x0 ax + bx + c ax + b b 1 0 a a ( 0 1) ; if b 0 ac 0 (mod ) 0 a ; if b 0 ac 6 0 (mod ) Thee have bee ome ae o he eimaio of d i 111 i ( 0 ; 111; 01 ) bu o igh boud i kow. The geeal lowe ad ue boud 6 (3 + ) o d i 111 i ( 0 ; 111; 01 ) wa develoed by Peala [8], which will be efeed o a Peala boud. Peala eaed zeo a a quadaic eidue, while we ake i a a oeidue. Thee wo kid of eame make lile diffeece. We ow deive a bee boud o d i 111 i ( 0 ; 1 ; 111; 01 ) To hi ed, we eed he calculaio ad eimaio of he chaace um 01 (a 1 ; 111;a ) x0 (x + a 1 ) 111 (x + a ) whee a 1; 111;a ae aiwie diic eleme of Z. The Peala boud ae baed maily o he followig eimae. Lemma 1 Le. The j (a 1 ; 111;a )j < ( 0 1) whee he a i ae aiwie diic eleme of Z. Thi eul i a ecial cae of a moe geeal eul abou mulilicaive chaace due o Weil [9, Theoem C,. 3]. Followig Peala [8], we call hi iequaliy he Weil boud.

3 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY (a 1 ;a ) ca be evaluaed exacly, ice i ca be educed o he calculaio of 1(a). Thi ca be geealized io he fac ha educe o if i eve, ad o (a 1 ; 111;a ) 01 (0; 1; b 1 ; 111;b 03 ) (0; 1; b 1 ; 111;b 0 ) if i odd. The oof of he cae wa give by Daveo [1]. We ow geealize Daveo oof fo he cae io he geeal cae of beig eve. Lemma Le be eve, ad le a 1; 111;a be aiwie diic eleme of Z. The (a 1;a ; 111;a )01 + c(a 1; 111;a ) 01(d ; 111;d ) The alyig he above afomaio yield (a 1; 111;a ) x60a y60hg y60hg y60hg F (x) F uy + v gy + h (gy + h) 0 i1 c y60hg i1 [(a ig + u)y + a ih + v] [(a i g + u)y + a i h + v] i (y + d i ) whee hee d i ae aiwie diic ad d i (a i 0 a )(a 1 0 a 3 ) (a i 0 a 1 )(a 0 a 3 ) ; i c 01 y0 i (y + d i) 0 1 c(a 1 ; 111;a )(a 1 0 a )(a 1 0 a 3 ) i [(a i 0 a 1 )(a 0 a 3 )] Poof The key o uch a educio i he ue of he afomaio x uy + v gy + h wih uh 0 gv 6 0(mod ), which give x6ug F (x) y60hg F uy + v gy + h whee F (x) i a olyomial of Z [x]. We chooe ow I ou cae, defie u 0 a 1(a 0 a 3) v 0a (a 1 0 a 3) g a 0 a 3 h a 1 0 a 3 F (x) (x + a 1) 111 (x + a ) 01(d; 111;d) 01 + c I ca be eaily ove ha d ; 111;d ae aiwie diic. We will efe o hi eul a he Daveo educio heoem, which will lay a imoa ole i develoig he ew boud. The followig combiaoial eul ae eeded i he equel. Thei oof ae adad ad ca be foud i may exbook o combiaoic. Lemma 3 Le. The 1) ) i eve i1 i i odd i 01 i (i 0 1) 01 ( 0 ) + 1. I he equel we aume 3. To deive he boud, we make ue of he fucio 1 01 k0 The followig lemma i eeded lae. Lemma Defie B(; ; i 0; 111;i 01) 1+ i +1 x 0 k 0k < 111 <k 01 whee ad ae oiive iege wih ad i 0; 111;i 01 f0; 1g. The oe ha (x )1fo all x 6 0 (mod ). We have jb(; ; i 0; 111;i 01)j c i (0hg + d i ) 1 1 oe ha x x 0 The followig cocluio i obviou.

4 1696 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY 1998 Lemma 5 Alyig Lemma 7 ad he Weil boud o he fomula above yield x0 k0 i x 0 k 0 x 0 + B(; ; i 0 ; 111;i 01 ) A we aume ha 0 ; 1 ; 111; 01 ae aiwie diic, he followig cocluio follow fom he ecod Jacobhal fomula decibed ealie. 01 Lemma 6 i +i x 0 k x0 0k <k 01 x 0 k 0B(; ;i 0; 111;i 01) 3 odd 3 1 ( 0 1) + ( 0 1) 0 ( 0 1) 0 1 ( 0 ) The followig lemma deive diecly fom he Daveo educio heoem. Lemma 7 Le 0 be eve. The 0k < 111 <k 01 (0 k ; 111; 0 k ) 0k < 111 <k 01 c( k ; 111; k ) 0 B(; ; i 0 ; 111;i 01 ) 01(0 0 k ; 111; 0 0 k ) whee c( k ; 111; k ) 6 0(mod ) ad ae defied i Lemma, ad whee k 0 ; 111;k 0 ae aiwie diic ad ae defied he ame a hoe d i i Lemma. By Lemma 5 ad 6 we obai ha x + 1 0k < 111 <k 01 (0 k ; 111; 0 k ) 0 B(; ;i 0 ; 111;i 01 ) + 3 0k < 111 <k 01 (0 k ; 111; 0 k ) 3 0k < 111 <k 01 odd (0 k ; 111; 0 k ) + 0k < 111 <k 01 (0 k ; 111; 0 k ) + 0 B(; ;i 0 ; 111;i 01 ) 1 ( 0 1) 0 0 ( 01 ( 0 ) ) ( 01 ( 0 3) + ) By Lemma B(; ; i 0; 111;i 01) 01+ Combiig he above wo fomula yield whee T 0 T x T () ( 01 ( 0 3) + ) ow le WH (i i 01) deoe he Hammig weigh of veco (i i 01 ), i.e., he umbe of 1 i he biay veco. Aume ha WH (i i 01) 0. oe ha j C i + j if ad oly if i j 0, o hee ae a mo eleme i f 0; 1; 111; 01g which belog o he e D i 111 i ( 0 ; 111; 01 ), whee. We ow aume ha hee ae exacly u uch eleme, ay 0; 1; 111; u01, whee u. Le ad The we have xa xb I follow ha B f 0 ; 1 ; 111; u01 g A D i 111 i ( 0 ; 111; 01 )B d i 111 i ( 0 ; 111; 01 ) 0 u u d i 111 i ( 0 ; 111; 01 ) 0 xd d i 111 i ( 0; 111; 01) xd

5 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY whee D i 111 i deoe D i 111 i ( 0 ; 111; 01 ). Hece, fo each (i i 01 ) we have d i 111 i ( 0; 111; 01) 0 xd d i 111 i ( 0 ; 111; 01 ) xd If he Hammig diace bewee he veco (j j 01) ad (i i 01) i geae ha o equal o wo, i i o difficul o ee xd ( ; 111; ) 0 Fuhemoe, if he Hammig diace bewee he wo veco (j j 01) ad (i i 01) i oe, i i clea ha oe ha i ; 111;i I follow ha 0 xd ( ; 111; ) 1 D i 111 i ( 0 ; 111; 01 )Z (3) f0; 1g d i 111 i ( 0 ; 111; 01 ) 0 xz d i 111 i ( 0; 111; 01)+ xz Combiig () ad () ove he followig eul. Pooiio Le he ymbol ad aumio be a befoe ad 3. We have 0W d i 111 i ( 0; 1; 111; 01) 0 W whee 0; 1; 111; 01 ae aiwie diic eleme of Z ad W ( 01 ( 0 3) + ) + 01 ( +1)0 1 oe ha W i eeially [( 0 3)] + ( +1). Ou ew boud ae much ueio o he Peala boud. The mai coibuio o he ew boud come fom he Daveo educio heoem ad eveal echique. Fo he cae 3, he above oof how ha d i i i ( 0 ; 1 ; ) umeical comuaio how ha hee lowe ad ue boud fo 3ae quie igh. I ca be ee fom he develome of he boud ha he ew boud ae uually igh fo mall. Pooiio how ha Legede equece have a ideal diibuio of ae of legh whe i mall. () I ummay, Legede equece have oimal balace bewee 0 ad 1, a lage liea a, he oimal auocoelaio oey, ad a ideal diibuio of ae of mall legh. III. PATTER DISTRIBUTIO AD WEIGHT DISTRIBUTIO I hi ecio we how a cloe elaio bewee he ae diibuio of Legede equece ad he weigh diibuio of Q.R. code. Thi i alo ue fo ome ohe equece ad code. Thu he imoace of he ae diibuio of equece i codig heoy follow. Le 61(mod8), ad le Q ad P deoe he e of quadaic eidue ad ha of quadaic oeidue modulo, eecively. The biay Q.R. code ae hoe wih idemoe Q x i P x i 1+ Q x i 1+ oe ha he idemoe Q xi above i he olyomial fom of he Legede equece defied i hi coeodece. I he equel we ue C o deoe he biay Q.R. code wih hi idemoe. The liea a of a equece i defied o be he legh of he hoe liea-feedback hif-egie ha oduce he equece [], [6], [7]. I i kow ha he Legede equece l 1 ha liea a ( +1) if 01 (mod 8) ad ( 01) if 1 (mod 8) [3]. Le l 0 l l 01 be he fi eiodic egme of he Legede equece, ad le L i deoe he i-e lef-hifed veio of he veco (l 0 ;l 1 ; 111;l 01 ) fo i 0; 1; 111;01. Sice l 1 ha liea a h, by he defiiio of liea a he veco L 0; 111;L h01 of GF () fom a bai of he Q.R. code C, whee h i eihe ( 01) o ( +1). Thu we have eached he followig cocluio. Pooiio 3 The Q.R. code C ha geeao maix G L 0 L 1.. L h01 whee L i ad h ae he ame a befoe. Thi eul, hough aighfowad, make a coecio bewee he Q.R. code ad he Legede equece. I allow u o eablih he elaiohi bewee he weigh diibuio of Q.R. code ad he ae diibuio of Legede equece. Le h deoe he liea a of he Legede equece. By Pooiio 3 evey codewod c of C i exeed a c L + L L whee 0 0 < 111 < 01 h 0 1. The Hammig weigh of c ad he ae diibuio of he Legede equece ae elaed a follow. Pooiio WH (c) i +111+i 1 P x i d i 111 i (0 0 ; 111; 0 01 ) (5) whee he addiio i i i 01 1i iege addiio modulo ad 0 i 0 i. Poof Le 0 ; 111; 01 be fixed ad he ame a befoe. Coide he e of (1). I i eay o ee ha D i 111 i ( 0 ; 111; 01 ) \ D j 111 j ( 0 ; 111; 01 );

6 1698 IEEE TRASACTIOS O IFORMATIO THEORY, VOL., O., JULY 1998 whee (j 0 ;j 1 ; 111;j 01) GF () ad (i 0 ;i 1 ; 111;i 01) ae diffee biay veco. The by (3) ad defiiio WH (c) i Z 01 j0 i +111+i 1 l +i 1 D i 111 i (0 0 ; 111; 0 01) jd i 111 i (0 0 ; 111; 0 01)j i +111+i 1 d i 111 i (0 0 ; 111; 0 01) i +111+i 1 oe ha hee d i 111 i ( 0 ; 111; 01) ae acually he umbe of ae ha aea i a cycle of he Legede equece. By Pooiio, he weigh diibuio of he Q.R. code i deemied if he ae diibuio of he Legede equece i deemied. Howeve, he covee may o be ue. I fac, (5) idicae ha he deemiaio of he ae diibuio of he Legede equece i much hade ha he weigh diibuio oblem of he Q.R. code. oe ha d i i ( 0; 1) ae give by Pooiio 1. Wih he hel of Pooiio hee d i i ( 0 ; 1 ) ca be ued o deemie ome weigh of he Q.R. code C. Fo 3 hee ae o kow fomula fo d i 111i ( 0; 111; 01), hough may well-kow mahemaicia have aacked hi oblem fo huded of yea. While calculaig d i 111 i ( 0 ; 111; 01) i had, we may hoe o develo igh boud o hem. Such igh boud ca be ued o ay omehig abou he weigh diibuio of Q.R. code. To illuae hi, we ake ou boud of Pooiio a a examle. Pooiio 5 The Q.R. code C ha a lea h 0 01 W WH (c) +01 W whee W ad h ae he ame a befoe. Poof Thee ae h e of iege 0 0 < 111< 01 h 0 1 Each e of uch iege defie a codewod c L L codewod c wih whee L i ae he ame a befoe, ad diffee e defie diffee codewod by he defiiio of liea a. oe ha x 0 + x x 01 1 ove GF ()[x 0 ; 111;x 01] ha 01 oluio (x 0; 111;x 01). The cocluio he follow fom Pooiio ad. The boud of Pooiio 5 ae ueful oly whe i mall. Whe h 3, i ay ha C ha a lea codewod c wih If we had bee boud WH (c) d i 111 i ( 0; 111; 01) 1 he oof of Pooiio 5 would alo ove ha hee ae a lea h codewod c wih Remak oe ha he above obevaio alo alie o ome ohe biay cyclic code ad biay equece. Le w 1 be a biay equece of eiod (o eceaily he lea eiod). Le h deoe he liea a of hi equece. The all he hifed veio of he fi eiodic egme of hi equece geeae a [; h] cyclic code. If he ae diibuio of hi equece i kow, he weigh diibuio of he cyclic code i kow, a Pooiio i alo ue i hi geeal cae. Hece, evey biay eiodic equece wih kow ae diibuio give a biay cyclic code wih kow weigh diibuio. Helleeh ha exeed he weigh of codewod of Q.R. code i em of Legede um [5], while we do i i em of he umbe of ae of ceai legh ee i a cycle of he Legede equece. I i oible o bidge he wo aoache o ge ome ieeig eul. IV. COCLUSIOS We have develoed ew boud o he umbe of ae aeaig i a cycle of Legede equece, which i much ueio o kow boud. We have alo howed a alicaio of uch boud i Q.R. code. We have exeed he weigh of codewod of a cla of biay cyclic code i em of he umbe of ae of ceai legh ee i a cycle of he equece. Thu he ae diibuio oblem i alo ieeig i codig heoy. ACKOWLEDGMET The auho wihe o hak he wo aoymou efeee fo oiig ou ome eo i he oigial veio of hi coeodece ad makig a umbe of uggeio ad comme ha gealy imoved he eeaio. REFERECES [1] H. Daveo, O he diibuio of quadaic eidue (mod ), J. Lodo Mah. Soc., vol. 6,. 9 5, [] C. Dig, G. Xiao, ad W. Sha, The Sabiliy Theoy of Seam Cihe. Heidelbeg, Gemay Sige-Velag, [3] C. Dig, T. Helleeh, ad W. Sha, O he liea comlexiy of Legede equece, IEEE Ta. Ifom. Theoy, vol., , May [] S. W. Golomb, Shif Regie Sequece. Lagua Hill, CA Aegea Pak, 198. [5] T. Helleeh, Legede um ad code elaed o Q.R. code, Dic. Al. Mah., vol. 35, , 199. [6] T. Helleeh ad P. V. Kuma, Sequece wih low coelaio, i Hadbook of Codig Theoy, Ple, Bualdi, ad Huffma, Ed. Amedam, The ehelad Elevie, [7] E. L. Key, A aalyi of he ucue ad comlexiy of oliea biay equece geeao, IEEE Ta. Ifom. Theoy, vol. IT-, , [8] R. Peala, O he diibuio of quadaic eidue ad oeidue modulo a ime umbe, Mah. Com., vol. 58,. 33 0, 199. [9] W. M. Schmid, Equaio ove Fiie Field A Elemeay Aoach (Lecue oe i Mahemaic, vol. 536). Beli, Gemay Sige- Velag, [10] D. V. Sawae ad M. B. Puley, Cocoelaio oeie of eudoadom ad elaed equece, Poc. IEEE, vol. 68, , May [11] T. Soe, Cycloomy ad Diffeece Se. Chicago, IL Maham, WH (c) 01 1 Theefoe, ay imoveme o he boud o d i 111 i ( 0; 111; 01) (6) would ehace ou kowledge abou Q.R. code.