Linear Complexity Over and Trace Representation of Lempel Cohn Eastman Sequences

Transcription

1 548 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 003 Lner Coplexty Over nd Tre Representton of Lepel Cohn Estn Sequenes Tor Helleseth, Fellow, IEEE, Sng-Hyo K, Student Meer, IEEE, nd Jong-Seon No, Meer, IEEE Astrt In ths orrespondene, the lner oplexty over of Lepel Cohn Estn (LCE) sequenes of perod for n odd pre s deterned. For 3 5 nd 7, the ext losed-for expressons for the lner oplexty over of LCE sequenes of perod re derved. Further, the tre representtons for LCE sequenes of perod for 3nd 5 re found y oputng the vlues of ll Fourer oeffents n for the sequenes. Index Ters Lepel Cohn Estn (LCE) sequenes, lner oplexty, sequenes. I. INTRODUCTION Aong propertes of perod sequenes [], [8], the lner oplexty [5], [6], [0], [4], lne, nd orrelton propertes re portnt for the pplton of stre phers nd ode-dvson ultple-ess (CDMA) ounton systes []. A nry sequene s sd to hve the lne property f the dfferene etween the nuer of s nd 0 s n perod of the sequene s t ost one. Let s(t) e nry sequene of perod n. The utoorrelton funton of nry sequene of perod n s defned s (0) s(t)s(t ) : A sequene s defned to hve del utoorrelton f n; f 0odn 0; otherwse. A lot of ttenton [7], [8], [7], [9] hs een devoted to nry sequenes of perod 0 wth del utoorrelton. A nry sequene of even perod n wth the lne property s sd to hve optl utoorrelton f 0 or 04; f n 0od4 or 0; f n od4. Let p e pre nd e postve nteger. Let F p e the fnte feld wth p eleents nd Fp 3 F p nf0g. Let S e nonepty suset of Fp 3 nd prtve eleent of F p. Then the hrterst sequene of perod p 0 of the set S s defned s [9] ; f t S 0; otherwse. Let S e set defned s [9], [] S 0 0 p 0 0 where p s n odd pre nd s prtve eleent of F p. Then, the hrterst sequene of ths set S s referred to s () Lepel Cohn Estn (LCE) sequene [], [], whh s 0- nry sequene of perod p 0,.e., of even length. It hs een shown tht LCE sequenes hve the optl utoorrelton nd lne property. No et l.[5] lso ntrodued nry sequenes of perod p 0 wth optl utoorrelton property y usng the ge of the polynol (z ) d z d over F p, whh turned out to e LCE sequenes. Let (x) denote the qudrt hrter of x defned y (x) ; f x s qudrt resdue 0; f x 0 0; f x s qudrt nonresdue. Helleseth nd Yng [9] desred LCE sequenes y usng the ndtor funton nd the qudrt hrter gven y () ( 0 I(t )0 ( t )) (3) where the ndtor funton I(x) f x 0nd I(x) 0otherwse. Helleseth nd Yng [9] studed the lner oplexty over F of LCE sequenes. Even though LCE sequenes re nry sequenes, they re onstruted sed on the fnte feld F p nd, thus, t s ore nturl to fnd the lner oplexty over F p for LCE sequenes. The tre representton of sequenes s useful for pleentng the genertor of sequenes nd nlyzng ther propertes [6], [], [8]. Thus, t s of gret nterest to represent LCE sequenes y usng the tre funtons. In ths orrespondene, the lner oplexty over F p of LCE sequenes of perod p 0 for n odd pre p s deterned. For p 3; 5; nd 7; the ext losed-for expressons for the lner oplexty over F p of LCE sequenes of perod p 0 re derved. Further, the tre representtons for LCE sequenes of perod p 0 for p 3 nd 5 re found y oputng the vlues of ll Fourer oeffents n F p for the sequenes. II. LINEAR COMPLEXITY OVER F p OF LCE SEQUENCES OF PERIOD p 0 It s well known tht the Fourer trnsfor of p-ry sequene s(t) of perod n p 0 n the fnte feld F p s gven s A n nd ts nverse Fourer trnsfor s 0 s(t) 0t (4) A t (5) where s prtve eleent of F p nd A F p. Usng the Fourer trnsfor of the sequenes, we frst fnd n expresson for A 0, 0 n 0 of LCE sequenes s n the followng le. Le : Let the p-d expnson of e gven s Mnusrpt reeved August 4, 00; revsed Jnury, 003. Ths work ws supported n prt y BK, ITRC, nd The Norwegn Reserh Counl. T. Helleseth s wth the Deprtent of Inforts, Unversty of Bergen, N-500 Bergen, Norwy (e-l: Tor.Helleseth@.u.no). S.-H. K nd J.-S. No re wth the Shool of Eletrl Engneerng nd Coputer Sene, Seoul Ntonl Unversty, Seoul, Kore (e-l: ksh@l.snu..kr; jsno@snu..kr). Counted y A. M. Klpper, Assote Edtor for Sequenes. Dgtl Ojet Identfer 0.09/TIT /03$ IEEE p (6) where 0 p 0. Then, A 0 of the LCE sequenes defned n (3) s gven s (p0)a 0 0(0) (0) 0 p 0 od p: (7)

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE Proof: Usng the Fourer trnsfor of the sequenes n (4), the relton for A0 n e derved s follows: na0 s(t) t ( 0 I( t )0 ( t )) t t 0 (0) 0 ( t ) t : (8) For 0, (8) n e gven s n na0 p ( t ) n p 0 0 ( t )() p odp: Thus, we hve proved tht the le holds for 0. For nonzero, (8) n e rewrtten s na0 0(0) 0 0(0) 0 xf yf (x )x 0 () (y)(y 0 ) : (9) As z vres over F p, z tkes ll the qudrt resdues n F p extly twe nd the zero eleent one. Slrly, z tkes ll the qudrt nonresdues n F p s vlues extly twe nd the zero eleent one. It s ler tht ll the qudrt resdues nd nonresdues together wth the eleent 0 over ll eleents n F p. Usng the defnton of the qudrt hrter () n (), (9) s odfed s na 0 0(0) 0 zf [(z )(z 0 ) (z )(z 0 ) ] 0(0) 0 zf [(z 0 ) 0 (z 0 ) ] 0(0) 0 l0 l (0) 0l ( 0 l ) zf z l : The nner su only ontrutes when l p 0, n ths se l 0. Note tht when l 0then 0 l 0. Therefore, we otn na 0 0(0) 0 (p 0 ) p 0 Redung odulo p for oth sdes, we hve the relton (p 0 )A 0 0(0) p 0 Fro the result of Lus [] gven y (0) redues to (7). p 0 p 0 (0) 0 : (0) 0 od p: (0) od p It s lredy known fro Blhut s theore [3], [4] tht the lner oplexty of perod sequenes n e deterned y oputng the Hng weght of ther Fourer trnsfor. Thus, we need to deterne the rdnlty of the set f j A 0 60; 0 n 0 g, whh s lulted fro (7). We hve proved the followng result. Theore : Let C e the nuer of ntegers, 0 p 0 stsfyng the relton p 0 (0) od p () where the s re oeffents n the p-d expnson p of. Then the lner oplexty over F p of the LCE sequene of perod n p 0 defned n (3) equls L p n 0 C: () To deonstrte ths tehnque, we wll lulte the lner oplexty over F p of the LCE sequene of perod n p 0 n the se of p 3; 5; nd 7. But t s not esy to fnd the lner oplexty over F p of LCE sequenes for p>7. A. Lner Coplexty Over F3 of LCE Sequenes of Perod 3 0 Usng the result of Theore, the lner oplexty over F3 of LCE sequenes of perod n 3 0 s derved n the followng theore. Theore 3: The lner oplexty over F3 of the LCE sequene of perod n 3 0 s gven s nd L3 3 0 : Proof: For p 3, t s ler tht 0 0; 3 0 ; even; f even odd; f odd. Then () s rewrtten s (0) od 3: (3) Thus, ll the s n the 3-d expnson of should e or. The nuer of solutons of ths syste s sne seletng 0; ;...; 0 unquely deternes. However, even though t stsfes (3), the soluton orrespondng to 0 0 ust e exluded sne t orresponds to 3 0. We onlude tht C 0 nd the lner oplexty over F3 of the LCE sequene of perod 3 0 equls L3 3 0 : B. Lner Coplexty Over F5 of LCE Sequenes of Perod 5 0 In ths se, the lner oplexty over F5 of LCE sequenes of perod 5 0 s derved y ountng nonzero Fourer oeffents of the sequenes s n the followng theore. Theore 4: The lner oplexty over F5 of the LCE sequene of perod n 5 0 s gven s L where d s the lrgest nteger less thn or equl to d.

3 550 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 003 Proof: Sne 5 0 s n even nteger for ny nteger, () for p 5n e rewrtten s od5 (4) where the s re oeffents n the 5-d expnson 5 of, 0 5 0, nd F5. It n e esly derved tht 0 0; 0; ; 3 3; 4 od 5: In order to stsfy (4), ll the s re lrger thn or equl to for 0 0 nd the nuer of ourrenes 3n the 5-d expnson of should e ultple of 4 euse the order of eleent 3 n F5 s 4, tht s, 3 4 od5. Tht s, 3ours tes nd or 4 ours 0 tes n the 5-d expnson of. For 0 5 0, the nuer of ntegers stsfyng (4) n e ounted s C 0 0 where 5 0 (4; 4; 4;...; 4) s exluded even though the nuer of ourrenes 3n the 5-d expnson of s 0od4, euse >5 0. Therefore, the lner oplexty over F5 of the LCE sequene of perod 5 0 s gven s L C : C. Lner Coplexty Over F7 of LCE Sequenes of Perod 7 0 Slrly to the prevous two ses of p 3nd 5, the lner oplexty over F7 of the LCE sequene of perod 7 0 s derved y ountng nonzero Fourer oeffents of the sequenes s n the followng theore. Theore 5: The lner oplexty over F7 of the LCE sequene of perod n 7 0 s gven s L u0 w0 v0 u ; 3v j; 6w k; D where d s the lrgest nteger less thn or equl to d nd D 0 u 0 0 3v 0 j 0 6w 0 k nd k; 0 k 5 s postve nteger stsfyng 0 od 6; f s even 3 k 3 od 6; f s odd. Proof: Usng the relton 7 0 even; f s even odd; f s odd () for p 7n e expressed s (5) (0) od 7 (6) where F7. Usng (6), the theore n e proved n slr nner to tht of the prevous theore. III. TRACE REPRESENTATION OF LCE SEQUENCES OF PERIOD p 0 In ths seton, the tre representton of LCE sequenes of perod p 0 for p 3nd 5 s derved y usng the tre funtons fro F p to F p, where kj, even though they re nry sequenes. For our sequenes, the A s n (5) re n F p. If the Fourer oeffents A s for ll eleents n oset orrespondng to the eleent hve the se vlue, then the suton of ll eleents n the oset kes the tre funton A tr( t ). Further, f A s hve the se vlues for ll eleents wthn the se osets of F p, (5) n e expressed s lner onton of the tre funtons over F p gven y A tr k (t ) (7) L where L s set of oset leders for the set of yloto osets odulo p 0, nd for eh L, F p s the sllest sufeld of F p ontnng. Thus, t s enough to fnd the Fourer oeffents A s for ll oset leders for the set of yloto osets odulo p 0 f A s hve the se vlues for ll eleents wthn the se oset. Let (0; ; ;...; ) e vetor orrespondng to the oeffents n the p-d expnson p of ; 0 p 0. It s ler tht ll ntegers orrespondng to the yl shft of vetor (0; ; ;...; ) elong to the se yloto oset of F p. The tre representton of the sequenes of perod p 0 s derved y oputng ll the A oeffents, 0 p 0 n (7) for the LCE sequenes n (3). A. Tre Representton of LCE Sequenes of Perod 3 0 In order to fnd the tre representton of LCE sequenes of perod 3 0, let e prtve eleent of the fnte feld F3. Let tr k (t ) denote the tre funton fro F3 to F3, where k j nd F3 s the sllest sufeld of F3 suh tht F3. We n lssfy the oset leders for the set of yloto osets odulo 3 0 s follows. I o : Set of odd oset leders, where every dgt n the 3-d expnson of oset leder only tkes the vlues or 0; for exple, 339(; ; ). I e : Set of even oset leders exludng the oset leder 0, where every dgt n the 3-d expnson of oset leder only tkes the vlues or 0; for exple, 0 9 (; 0; ). I o : Set of odd oset leders nludng I o. I e : Set of even oset leders nludng I e. Usng the ove notton, the tre representton of the LCE sequene of perod 3 0 s gven n the followng theore. Theore 6: The tre representton of the LCE sequene of perod n 3 0 s gven y I ni tr k ( t ) I ni tr k ( t ) I tr k ( t ): Proof: For the LCE sequenes of perod 3 0, the oeffents A F3, defned n (7) n e rewrtten s A 0 0(0) (0) 0 od 3: (8) Now, we hve to fnd ll A s, for the tre representton of the LCE sequenes of perod 3 0.For 0, t s esy

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE to fnd tht A0. Clerly, for odd, 3 0 odnd for even, 3 0 0od. Then (8) n e odfed s follows: (A0 (0) )(0) 0 od 3: (9) Note tht j 0 n 0, j 3 0, where A0 for j 0s lredy found. In the 3-d expnson of 3 nd j j 3, t s ler tht j p for ll, 0 0. Let us onsder three ses s follows. Cse : A0 A j 0: We hve to fnd ll j n 0, j 3 0 suh tht A0 0 n (9), whh s rewrtten s (0) od 3: (0) A neessry ondton for (0) s tht the s n the 3-d expnson 3 of only tke the vlues or, whh ens tht the j s only tke the vlues 0 or. Sne 0 od3nd od3, the nuer of ourrenes, 0 0 n the 3-d expnson of stsfyng (0) should e odd for odd nd even for even nd, thus, the nuer of ourrenes should e even for ny nteger. Therefore, the nuer of ourrenes of n the lst of j ; 0 0 should e even for ny nteger nd, thus, j s even. Therefore, the oset leder of j suh tht A j 0elongs to the set I e, where j 0s exluded. Cse : A0 A j : In ths se, we hve to fnd ll j n 0 ; j 3 0 suh tht A j n (9). The followng two suses re onsdered. ) Cse of even nteger (.e., j even nteger): We n rewrte (9) s 0(0) od 3 () where ll s n the 3-d expnson of hve to tke the vlues or. The nuer of ourrenes n the 3-d expnson of should e odd for even nd even for odd, whh ens tht the nuer of ourrenes, 0 0 n the 3-d expnson of should e odd for ny nteger. Therefore, ll j s only tke the vlue 0 or nd the nuer of ourrenes of j n the 3-d expnson of j should e odd for ny nteger, whh ens tht j s odd. Ths ontrdts the ssupton tht j s n even nteger. Therefore, there s no even nteger j whh kes A j. ) Cse of odd nteger (.e., j odd nteger): Equton (9) n e wrtten s 0 od3: () Equton () ens tht t lest one of s n the 3-d expnson of hs to tke the vlue 0, whh ens tht t lest one of j s n the 3-d expnson of j hs to tke the vlue. Therefore, the oset leder of j elongs to the set I o ni o. Cse 3: A0 A j : In ths se, ll j n 0, j 3 0 suh tht A j n (9), hve to e deterned, whh n e esly found euse we hve lredy found ll j s suh tht A j 0or. Clerly, the renng sets of oset leders for the set of yloto osets odulo 3 0 re I e ni e nd I o. For p 3, the tre representton for LCE sequene of perod 80 s gven n the followng exple, where the tre funton s defned n Theore 6. Exple 7: For n nd 4, the LCE sequene s(t) of perod 80 s otned s : The oset leders for the set of yloto osets odulo n e lssfed s I o f; 3g I e f4; 0; 40g I o ni o f5; 7; ; 7; 3; 5; 4; 53g I e ni e f0; ; 8; 4; 6; 0; ; 6; 44; 50g: Then the LCE sequene s(t) of perod 80 n e expressed s lner onton of tre funtons over F3 s follows: ftr( 4 5t )tr( 4 7t )tr( 4 t )tr( 4 7t ) tr( 4 3t )tr( 4 5t )tr( 4 4t )tr( 4 53t )g ftr( 0t )tr( 4 t )tr( 4 8t )tr( 4 4t ) tr( 4 6t )tr( 0t )tr( 4 t )tr( 4 6t ) tr( 4 44t )tr( 50t )g ftr( 4 t )tr( 4 3t )g where s prtve eleent of F3. B. Tre Representton of LCE Sequenes of Perod 5 0 For the perod 5 0, the tre representton of LCE sequenes s derved slrly to the se of perod 3 0. Let e prtve eleent of the fnte feld F5. Let tr k (t ) denote the tre funton fro F5 to F5, where k j nd F 5 s the sllest sufeld of F5 suh tht F5. The oset leders for the set of yloto osets odulo 5 0 n e lssfed s follows. I o : Set of odd oset leders, where every dgt n the 5-d expnson of oset leder only tkes the vlues 0,, or nd the nuer of ourrenes of n the 5-d expnson of oset leder s od4. I o 3 : Set of odd oset leders, where every dgt n the 5-d expnson of oset leder only tkes the vlues 0,, or nd the nuer of ourrenes of n the 5-d expnson of oset leder s 3od4. I e 0 : Set of even oset leders exludng oset leder 0, where every dgt n the 5-d expnson of oset leder only tkes the vlues 0,,or nd the nuer of ourrenes of n the 5-d expnson of oset leder s 0od4. I e : Set of even oset leders, where every dgt n the 5-d expnson of oset leder only tkes the vlues 0,, or nd the nuer of ourrenes of n the 5-d expnson of oset leder s od4. I o : Set of odd oset leders nludng I o nd I o 3. I e : Set of even oset leders nludng I e 0 nd I o. Usng the preedng notton, the tre representton of LCE sequene of perod 5 0 s gven n the followng theore.

5 55 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 6, JUNE 003 Theore 8: The tre representton of LCE sequene of perod n 5 0 s gven s I nfi [I g I nfi [I g I I tr k ( t ) tr k ( t ) tr k ( t ): 3 tr k ( t ) I 3 tr k ( t ) Proof: For the LCE sequenes of perod 5 0, the oeffents A F5, defned n (7) n e rewrtten s 3A 0 0(0) (0) 0 od 5: (3) Usng (3), the theore n e proved n the se nner s n the prevous theore. For p 5, the tre representton for LCE sequene of perod 4 s gven n the followng exple, where the tre funton s defned n Theore 8. Exple 9: For n nd 3, the LCE sequene s(t) s gven s : The oset leders for the set of yloto osets odulo n e lssfed s follows: I o f; 7; ; 37g I3 o f3g I0 e f; ; 6g I e f6; 3g I o nfi o [ I3 o g f3; 9; 3; 7; 9; ; 3; 33; 39; 43; 47; 49; 63; 69; 73; 93; 99g I e nfi e [ I3 e g f0; 4; 8; 4; 6; 8; ; 4; 34; 38; 4; 44; 48; 64; 68; 74; 94g: Then, the LCE sequene s(t) of perod 4 n e expressed s lner onton of tre funtons over F5 s follows: I nfi [I g I nfi [I g tr k ( t ) 3 tr k ( t ) REFERENCES [] L. D. Buert, Cyl Dfferene Sets (Leture Notes n Mthets). New York: Sprnger-Verlg, 97. [] E. R. Berlekp, Alger Codng Theory, revsed ed. Lgun Hlls, CA: Aegen Prk, 984. [3] R. E. Blhut, Trnsfor tehnques for error ontrol odes, IBM J. Res. Develop., vol. 3, pp , 979. [4], Theory nd Prte of Error Control Codes. New York: Addson-Wesley, 983. [5] H. Chung nd J.-S. No, Lner spn of extended sequenes nd sded GMW sequenes, IEEE Trns. Infor. Theory, vol. 45, pp , Sept [6] C. Dng, T. Helleseth, nd W. Shn, On the lner oplexty of Legendre sequenes, IEEE Trns. Infor. Theory, vol. 44, pp , My 998. [7] H. Doertn, Ks power funtons, perutton polynols nd yl dfferene sets, n Pro. NATO Advned Study Insttute Workshop on Dfferene Sets, Sequenes nd Ther Correlton Propertes,, Bd Wndshe, Gerny, Aug. 3 4, 998. [8] S. W. Golo, Shft-Regster Sequenes. Sn Frnso/Lgun Hlls, CA: Holden- Dy/Aegen Prk, 967/98. [9] T. Helleseth nd K. Yng, On nry sequenes of perod wth optl utoorrelton, n Pro. 00 Conf. Sequenes nd Ther Appltons (SETA 0), Bergen, Norwy, My 3 7, 00, pp [0] D. Jungnkel, Fnte Felds. Mnnhe, Gerny: B. I. Wssenshftsverlg, 993. [] J.-H. K nd H.-Y. Song, Chrterst polynol nd lner oplexty of Hll s sext resdue sequenes, n Pro. 00 Conf. Sequenes nd Ther Appltons (SETA 0), Bergen, Norwy, My 3 7, 00, pp [] A. Lepel, M. Cohn, nd W. L. Estn, A lss of nry sequenes wth optl utoorrelton propertes, IEEE Trns. Infor. Theory, vol. IT 3, pp. 38 4, Jn [3] R. Ldl nd H. Nederreter, Fnte felds, n Enyloped of Mthets nd Its Appltons. Redng, MA: Addson-Wesley, 983, vol. 0. [4] F. J. MWlls nd N. J. A. Slone, The Theory of Error-Corretng Codes. Asterd, The Netherlnds: North-Hollnd, 977. [5] J.-S. No, H. Chung, H.-Y. Song, K. Yng, J.-D. Lee, nd T. Helleseth, New onstruton for nry sequenes of perod wth optl utoorrelton usng ( ), IEEE Trns. Infor. Theory, vol. 47, pp , My 00. [6] J.-S. No, H. Chung, nd M.-S. Yun, Bnry pseudorndo sequenes of perod wth del utoorrelton generted y the polynol ( ), IEEE Trns. Infor. Theory, vol. 45, pp. 78 8, My 999. [7] J.-S. No, S. W. Golo, G. Gong, H.-K. Lee, nd P. Gl, Bnry pseudorndo sequenes of perod wth del utoorrelton, IEEE Trns. Infor. Theory, vol. 44, pp , Mr [8] J.-S. No, H.-K. Lee, H. Chung, H.-Y. Song, nd K. Yng, Tre representton of Legendre sequene of Mersenne pre perod, IEEE Trns. Infor. Theory, vol. 4, pp , Nov [9] J.-S. No, K. Yng, H. Chung, nd H.-Y. Song, On the onstruton of nry sequenes wth del utoorrelton property, n Pro. 996 IEEE Int. Syp. Inforton Theory nd Its Appltons (ISITA 96), Vtor, BC, Cnd, Sept. 7 0, 996, pp [0] R. A. Sholtz nd L. R. Welh, GMW sequenes, IEEE Trns. Infor. Theory, vol. IT 30, pp , My 984. [] V. M. Sdelnkov, Soe -vlued pseudo-rndo nd nerly equdstnt odes, Prol. Pered. Infor., vol. 5, no., pp. 6, 969. [] M. K. Son, J. K. Our, R. A. Sholtz, nd B. K. Levtt, Spred Spetru Countons, revsed ed. Rokvlle, MD/New York: Coputer Sene/MGrw-Hll, 985/994, vol.. [3] T. Storer, Cylotoy nd Dfferene Sets (Leture Notes n Advned Mthets). Chgo, IL: Mrkh, 967. [4] R. Turyn, The lner generton of the Legendre sequenes, J. So. Ind. Appl. Mth, vol., no., pp. 5 7, 964. I I tr k ( t ) tr k ( t ) I 3 tr k ( t ) where s prtve eleent of F5.