New families of p-ary sequences with low correlation and large linear span

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1 THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volu 4 Issu 4 Dcbr 7 TONG X WEN Qao-ya Nw fals of -ary sucs wth low corrlato ad larg lar sa CLC ubr TN98 Docut A Artcl ID ( Absact Ths artcl rsts a w faly of -ary sucs Th roosd sucs ar rovd to hav ot oly low corrlato rorty but also larg lar sa ad larg faly sz Furthror t shows that th w faly of sucs cotas Tag s cosucto as a subst f -sucs ar xcludd fro both cosuctos Kywords -ary sucs corrlato fucto faly sz lar sa uadratc for Ioducto A faly of sucs wth low corrlato ad larg lar sa has ortat alcatos cod dvso ultl accss (CDMA coucatos srad scu systs ad broadbad satllt coucatos [] Th sucs wth low corrlato usd CDMA coucatos ca succssfully cobat frc fro th othr usrs who shar a coo chal May fals of bary sucs of rod wth low corrlato hav b rortd Th Gold suc [] achvg th Sdlkov boud th larg ad sall fals of Kasa sucs [3] as wll as th GKW-lk sucs Rf [4] all hav dsrabl corrlato rorts Th cosuctos wr xtdd to th obary cas as kuar- oro (KM sucs Rf [5] Trachtbrg- Hllsth (TH sucs Rfs [6 7] ad TH-lk sucs Rf [8] Howvr ths sucs hav sall valus of lar sa I Rf [9] by rlaxg corrlato Yu ad Gog cosuctd a faly of suc wth largr lar sa ad faly sz I ths artcl Tag s cosucto Rf [8] s gralzd at th rc of th dcras of axu lar sa ad th cras of axu corrlato A w faly of -ary sucs S ( s cosuctd for wth odd ad Rcvd dat: TONG X ( WEN Qao-ya Stat ky Laboratory of Ntworkg ad Swtchg Tchology Bg Uvrsty of Posts ad Tlcoucatos Bg 876 Cha E-al: togx3@saco a tgr ( / Wh S ( s th faly of th TH-lk sucs cosuctd by Tag [8] Prlars Assu that s a odd r ad whr s odd Lt F b th ft fld wth lts lt for slcty dot F as F ad F as F Th th ac fucto ( fro F to th subfld F s dfd by ( x x Th ac fucto has th followg rorts: ( ax by a ( x b ( y ; for ab F x y F ; ( x ( x ; for x F Ay suc { s ( t } ovr F of rod has a ac rrstato that s thr xsts a fucto g( x fro F to F satsfs th followg codto k k Γ ( k g( x ( Ak x ; Ak F k t Such that st ( g( x F t Hr Γ ( s th st corsg of all cost ladrs odulo k k s th sz of th cyclotoc cost s a rtv lt of F Lt S b a faly of M -ary sucs of rod N gv by S { s ( t M t N } Th th corrlato fucto btw two sucs { s ( t } ad { s ( t } s dfd as C s s N s ( t s ( t ( ω ; M N ( t π / whr ω s th colx th root of uty gv by ω Th axu agtud of th corrlato C ax valus s dfd by C ax C ( ; M N ax s s

2 54 Th Joural of CHUPT 7 whr f Clarly Cax s th axu agtud of all oval auto- ad cross corrlato of th suc S Th st S s calld a (N M C ax faly of sucs whr N s th rod of th suc M s th faly sz ad s th axu corrlato agtud Th C ax suc faly has low corrlato f Cax c N whr c s a costat Th lar sa of a rodc suc s th lgth of th shortst lar fdback shft rgss that ca grat th suc I th followg th rlatoshs btw th rak of a uadratc for ( x ovr F ad th cross-corrlato fucto of th sucs wth th ac rrstato ( x ar show Lt a bass for x x whr x F ad F ovr F Th th fucto ( x s a uadratc for ovr F f t ca b xrssd as: x ( x b xx ; b F That s ( x s a hoogous olyoal of dgr rg F[ x x x ] Th rak of th uadratc for ( x s th u ubr of varabls rurd to rrst th fucto udr th osgular coordat asforatos It s rlatd to th dso of th vctor subsac W F that s W { ω x ( ω ( x for all x F } ( F Mor rcsly th rak r d W La (Hllsth-Gog [] If x ( s f( x s a uadratc for th th cross-corrlato fucto ( ca b wrtt as: Cs s( S( C s s ( ( x ( λ ( x whr S( ω ω λ s a osuar F La (Hllsth-Gog [] Lt ( x b a uadratc for ovr F of odd rak r ad λ s a osuar F th S( ω ω ( ( x ( λ ( x La 3 (Tag [8] Lt ( x b a uadratc for ovr F of v rak r ad λ s a osuar F th ( ( x ( λ ( x r/ S( ω ω ± Thor Lt gcd ( k ad / b a odd tgr If k d ( / whr d (od for ay < th cross-corrlato fuctos tak th followg thr valus: ( / ( / ts ts ( / ( / ts 3 Nw faly of -ary sucs wth larg sz Cosucto For wth a odd ad a tgr ( / lt S ( { s ( t } a faly of -ary sucs S ( s dfd by ( / ( l l t t / t( / l l l s ( t ( v ( v ( (3 La 4 All sucs S ( ar cyclcally dstct Thus th faly sz of S ( s Proof A t-shftd vrso of a suc S ( s rrstd as ( ( / ( l t t ( ( l l s t v v ( / l ( ( / ( l t For all t t s dtcal to th suc of E (3 f ad oly f t v v ( l v v / ; < (4 l l ad l ( / ; (5 l For odd sc gcd ( l ad 4 / l l th gcd ( ( / thus s th uu soluto E (5 whch oly gvs a val soluto of v l v l for l < Thus th sucs S ( for ay vl F wth l < ar cyclcally dstct I th followg th a thor of ths artcl s rovdd Thor For wth a odd ad a tgr ( / th faly S ( has cyclcally dstct -ary sucs of rod sucs s [ (43 ]/ [ (43 ]/ Th corrlato fucto of ( -valud ad axu corrlato s Thrfor faly of sucs ( S costtuts a ( Proof Th coutato of th corrlato fucto C ( btw two sucs { s ( t } ad { s ( t } s s ca b dvdd to four cass ddg o dffrt valus of ad Cas

3 No 4 TONG X t al: Nw fals of -ary sucs wth low corrlato ad larg lar sa 55 I ths val cas C ( s s Cas ad It s saghtforward that Cs s( { s ( t } s ust a -suc ovr F Cas 3 or sc th suc Th cross corrlato btw th suc s vstgatd ( / l l t t( / t( / l l l ( ( a ( v v ad th -suc b Fro La C a b t t ( ( / ( l / ( l t t t / t v vl l l ( ω ( ( x ( λ ( x ω ω whr λ s a osuar F ad th uadratc for ( / ( l l x vx l x ( λ x (6 l l For odd gcd ( hc t s dat that th st 4 ( { } odulo s uvalt to th ( st { } Sc (od for ( / ad ( d x ( th E (6 ca b rwrtt as d x ( / ( l l x vx l x ( λ x (7 l l To cout th rak of (x t s suffct to fd th ubr of th lts th vctor subsac W whch s dfd by E ( thus t s uvalt to fdg solutos to th followg uato ( ( l l x vl ω ω ( ω (( λ ω l ( ( ( / l l vl ω ω λ ω l l I fact ( / ( l l vl ω ω ( λ l l ( l l ω ( ( (( Thus ω vl ω ω ω λ ω l ( ( l A vl ω ω ( ω (( λ ω (8 l Obvously A ( A ( Sc th axu dgr of A 4( s th t has at ost 4( solutos For ( ω F th 4( total ubr of solutos of E (8 s at ost ( Thrfor th rak of uadratc for (x s (4 3 Fro Las 3 th axu corrlato [ (43 ]/ taks th valu Cas 4 Fro La ( ( x ( λ ( x Ca b( ω ω whr λ s a osuar F ad th uadratc for l ( l / l l l l l x ( ( v x ( v x ( / ( / l l l ( / x x vx v x l l (9 Sc s a odd tgr t s clar that a uadratc orsdu F s also a uadratc orsdu Thus th lt y F F γ y whr ad could b xrssd as F γ s or uadratc orsdus F Th slar to cas 3 (x ca b rwrtt as x ( ( v x ( v ( yx l l l γ l l l ( / ( / l l x γ ( yx vx γv ( yx l l ( To cout th rak of (x t s suffct to fd th ubr of th lts th vctor subsac W whch s dfd by E ( thus t s uvalt to fdg solutos to th followg uato l l l l x ( vl ( ω ω γ ( vl (( yω ( yω l l ( ( ( ( ω γ y yω v ω γ v y ω ( ( ( ( / l l l vl ω γ vl yω ω l l l ( / γ ( yω v ω γv y ω l l Slar to abov dscusso oly th followg d to b cosdrd: ( ( l l l l ( vl ω ω γ ( vl ( yω ( yω l l ( ω γy ( yω (v ωγ(v y ω Th lft-had sd of E ( s dotd as B Th: ( Sc th axu dgr of B B ( B 4( s ( th

4 56 Th Joural of CHUPT 7 4( t has at ost solutos If y F th total ubr of solutos of E( s at ost 4 3 wh ( ω rus through 4( (4 3 F If y F th total 4( ubr of solutos of E ( s at ost (4 4 wh ( ω ad ( yω ru through F Thrfor th rak of uadratc for (x s (4 3 ad (4 Fro Las ad 3 th axu corrlato taks th valu Rark If [ (43 ]/ th ( / t t l ( / l s( t ( v ( ( whch rrsts th sucs oducd by Tag for wth odd ad k cas Rf [8] Fro Thor th sucs gv by E ( hav four-valud corrlato ( / { ± } for all v F 4 Lar sa of th w faly S ( I ths scto th lar sa of th w roosd faly S ( s vstgatd Th lar sa of th suc { s ( t } ca b dd by xadg th xrsso of th suc { s ( t } as a olyoal t of dgr lss tha ad th coutg th ubr of ooals wth ozro coffcts occurrg th xaso Ths tchu wll b ald to d th lar sa of sucs th faly ( Thor 3 Th axu ad u la r sa of sucs S ( ar [ ( ]/ ad [ ( ]/ rsctvly Proof Fro th cosucto ad E (3 whch dots th ac rrstato of th suc s ( t S ( t s asy to s that E (3 has at ost of ( / ozro ac s ad ach ac has th lar sa of Thrfor th axu lar sa of sucs ( s gv by ( LS ax ( Slarly E (3 has at last of ( / ozro ac s ad th lar sa of ach ac s Th th u lar sa of sucs S ( s gv by LS 5 Coclusos ( ( For wth odd ad a tgr ( / a S S w -ary sucs faly S ( has b rstd ths artcl Th w faly S ( costtuts a ( [ (43 ]/ faly of sucs Coard w th th rvous cosuctos of sucs faly wth low corrlato [57] th w thods ca grat sucs faly that ot oly hav a largr faly sz but also a largr lar sa Wh th ara Tag's cosucto Rf [8] ca b rgardd as a subst of th w cosucto Tabl shows th coarso of th suc fals wth low corrlato Furthror th w sucs faly s flxbl that a ror valu of aras ad th corrsodg faly for a scfc alcato ca b chos Wh low corrlato s or crucal tha larg faly sz a sall valu of ad ca b chos Wh larg faly sz s or ortat a larg valu of ad ca b chos Tabl Coarso of th suc fals wth low corrlato Faly Faly sz C ax Lar sa Gold [] ( / Kasa [3] / 3 / G KW-lk [4] odd ( / ( / KM [5] odd TH [67] odd / ( / TH-lk [8] odd ( / ( / Nw faly odd [ (4 3 ]/ ( / Ackowldgts Ths work s suortd by th Maor Rsarch la of th Natoal Natural Scc Foudato of Cha (9643 th Natural Scc Foudato of Bg (47 th Natoal Rsarch Foudato for th Doctoral Progra of Hghr Educato of Cha (437 th Natoal Hgh Tchology Rsarch ad Dvlot Progra of Cha (6AAZ49 Rfrcs So M Oura J Scholtz R t al Srad scu coucatos hadbook Nw York: McGraw-Hll Ic Gold R Maxal rcursv sucs wth 3-valud rcursv crosscorrlato fuctos IEEE Trasactos o Iforato Thory 968 4(: Kasa T Th wght urators for svral classs of subcods of th d ordr Rd-Mullr cods Iforato ad Cool 97 8 (4: K S No J Nw fals of bary sucs wth low corrlato IEEE Trasactos o Iforato Thory 3 49(: Kuar P Moro O Pr-has sucs wth rodc corrlato rorts bt tha bary sucs IEEE Trasactos o Iforato Thory 99 37(3: Trachtbrg H O th crosscorrlato fuctos of axal lar rcurrg sucs Los Agls CA USA: Uvrsty of Southr Calfora 97

5 No 4 TONG X t al: Nw fals of -ary sucs wth low corrlato ad larg lar sa 57 To 9

The probability of Riemann's hypothesis being true is. equal to 1. Yuyang Zhu 1

The probability of Riemann's hypothesis being true is. equal to 1. Yuyang Zhu 1 Th robablty of Ra's hyothss bg tru s ual to Yuyag Zhu Abstract Lt P b th st of all r ubrs P b th -th ( ) lt of P ascdg ordr of sz b ostv tgrs ad s a rutato of wth Th followg rsults ar gv ths ar: () Th