1EICE TRANS Fllh'nAC1FWl'hI.N. VOI,.W2 A. YO I J!i.\vnnr rctl'l

342 I

On Some Properties of M-Ary Sidel'nikov Sequ~

dung-Sno CHUNG', Young-Sik K13ft, Tae-1Tyung LIM'"', IVanmenrhers, Jong-Seon NOfh', Member, and Habong CHUNG; :'', ,Wonmcmbcr

SIJhMARY In t h r q Icttcr. uec cnumcrnrc Ilic nunibcr of cycl ic~l ly in- etliiiiralc~lr M-:ul, Sidcl'nika~ scqurncrc ul'given length as well its the num- hcr oC diqtinci autocomlii~iun diqtri biitirms that they can l~ave, while rve c h : ~ n ~ e the prinfitive element for pncrnt ing the sctlucncc. k ~ y words: ~ ~ ~ f o c o r r ~ i ~ fio~r, o l t ~ ( j c ( > r t ~ / ( ~ t i o t I c / i \ ! ~ . i / ~ i ~ ~ i o r ; . ~ . <,v( \ O ~ ( J I T ~ ; C

~~rrrrihrt; I W - r ~ r y reqrrrrl.llr'Ps. .Yi(/~l'rlikn~~ s(Jqirrriren

1. Introduction

Like a quadrature phase qhift keying IfJPSK) niodula tinn for the sccond and thc third gencrittion wireless communi- calion s y s ~ c m ~ . M-ary phasc shift kcying (PSK) modulatiozl schemes ate fi-equeutly adopted as a st:uldard. Thus, it is in- teresting to find M-ary codcs with 2ood crror correctability as wcll as thc Tarnily of M-;IT sequences with good correla- tion property for a sivcn pncitive in t e ~ e r 1211. For these appli- cations, knowing the currelalion property of :he scqucilcc togctl~er with the nurnher of cycl ic:~ll y distinct sequences must be es~ential and important.

For ;I prime p, positive integers M and 37 such that MIp" - I. Sidel'ni kov I I ] conw-uctcd M-ary sequences (c;lllcd S i d ~ l ' n i h v . r e q l r n ~ r ~ s ) of period 11" - I , [he out-of- phase autc>correlation magnitude of which i s upper hnundcd by 4. Lelnpel, Cohn. and Ensman 121 inrrorluced the binary Sidel'ni kav sequences of period p" - 1.

Rcccnlly. Kim, Chung. No, and Chting 131 derived the autocorrelation di aributinn of M-ary Sidel'nikov sequences uqiny cyclotomic numbers of order M. It was also pointed out h a t the tcltal numhcl- o f clistinct nutocnrrelation values

elements of the corre5ponding ficld uqcd for gcncratine the scquencex are diffeset~t. Then naturally. the lbllowing tivo questions call be raised. "How many c ; ~ l icnlly i ncquivalcnt M-a1.y Sidel' ni!iov sequences arc thcrc?"'Arc the nutocor- rctation dirtributionc: of two cyclicnlly iuequivalcnt M-ary Sidel'nikuv sequences distinct'?" In this letter. a sequel lo

131. we ailswer these twu rlues~ions. By showing the reln tion between two ,Wary SideI'nikov sequenccc, cach from different priniitivc elcmcnrs. we enumerate the number of cyclically inequivrtlent M-ary Sidcl' nikov sequences a< tvcll as the nurnher of distinct ar~tocnrrelation distributiuns.

2. Preliminaries

Fnr an M-ary wquence s(r) of period N. the autocorrelation fiinction R(T) is defined as

where fil:~f = P J ~ ~ ~ ' ' ' .

Definition 1: [ I ] Let p be a prime and a a primitive eIe- ment in the finite field fi-,,. with g" elements. Let M he a positive integer such that M 2 2 and Mlp" - 1. Let St.

' k = 0, 1. . - .. M - I . bc thc disjoint suhretp of F," defined by

depends not only on M but also on thc period of the se- An M-ary Sidel'ni kov sequence s(t) of period p" - 1 i s rle- quelice, but always less than or equal to (:') + I .

In gcncral. two M - a ~ y Sidel ' nikov sequences of pe- tined ns

riod p" - I can be distinct whcn thc r.cspcctivc primitive k, i f a P ~ S n . 0 ~ k 1 ~ 1 . I - I s(r) =

M;~-u~script received dune 23. 2008. ko, i f ( rr=-I 0)

'The authArs are with tlie Dcpartmcnt of Elcclricnl Engineer- \

in? and Computer kierlct. :~nd 1h'MC. S e ~ u l Ni~tianzl Wnivcrsi~y, where ko is some integer modulo M Seoul 1 5 1-744. Korcn.

"The author i r with the Schonl of Electronics and Electrical 11 i s known that the M-ary Sidcl'n ikov scqucnce . ~ ( t ) in Enyinecring. Hongik University. Seoul 12 1 -79 1 . Korea. (3) can be rcpre~ented in terms of the multiplicative charnc-

'This research w a s supponcd by the MEST. the MKE, and thc ter @ M ( . ) 01' order M in F,. and the indicator. fi~nczion 1(+3 MOLAR, Knrea. ifirrwyh the fbhtering project of the Lahorntory as of EhceIlency and hy thc IT R&D prng1.ai1-t of MKE/I1TA 1200X-F- 007-01. Intelligent Wireless Cnmrnunic;ition Syste~ns i n 3 Ditncn- w':;' = w$l(af + 1) + rl.M(nr + 1) sional Environment].

(4)

a ) E-mail: {~nteger. kingsl. jnycll @ccl.snu.ac.kr Note that the indicmor filnctior~ i s defined as h) E-mnil: j~no(~'snu.ac.kr c ) E-mail: hribchung~~ 11orlyik.a~. kr 1 . i f . r=O

DOT: 10.158'I/tr'anxfun.E~2.A.342 I ( x ) = 0, if .r + 0

Copyright @ 2009 The Tnstitl~te of Electronics, Information and Communication Engineers

and the muItiplicarive character $J,~ is defined as I ~ ~ ~ ( U ' ) = e ~ ? n r / ~ and I J / ~ ( O ) = O in [4].

Ir i s ~hown in [3] that the autocorrelation distribution of a M-ary Sidel'nikov sequence can be represented by the cycJotomic numbers of order M defined below.

Definition 2: 151 Let cu be a primitive element of F,,, and M a positive integer such that M 2 2 2nd Mlp" - 1. The cyciocomic classes C,,, 0 I zf I M - 1, in F p are defined as

For fixed positive intepers 11 and v, not necessarily distinct. the cyclotomic number (u , ulb, is defined as the number of elements z r C,, such that 1 f z E C,,.

The following lemma lists some usefuL properties of cyclotomic numbers.

Lemma 1 : [ S ] 1) (i, jlM = [ M - i- j - iIM 2) ( i , j ) ~ = (pi, p j ) ~ 31 (i, j ) ~

( , j9 i ) n f . if is even or p = 2

+ i + i f % i s o d d i l i 1 l i ~ i 2 4) (i. . j I M r = Z:sl c:=(: (i + tM', j f sM')M

for A4 = rnM'.

Baumert, Mills. and Ward showed that the cyclotornic numbers of a certain order M over F,,. arc un~fnrm as fol- lows.

Theorem 1: 161 Let p be a prime. q = p2""" and M a divi- sor ot'p'+ I , such that M 1 3. Then the cycloton~ic numbers of order M over Fq are uniform as

(O,O)*f = v2 - (M - 3177 - 1

(0, iIM = (i, = ( i , i ) ~ = 77' + 7, for i f 0

, j = , otherwise

- 1 M if miseven where Q =

- - 1 , if misodd.

3. Counting the Cyclically Inerluivalent M-Ary Sidel'nikov Sequences

where gcd(c, p" - 1) = 1 and C-' is computed modulo M

Proofi Sincc s r ' = S,n . s r L ' ' ( A = k ilnplies that d' E

Srk. which further iinplies thal s ( t a t ) = cak. i.e., r.-'slrt) = k. The proof for the oppoqrte direction can he done in the exact same way. •

Using Lcmma 2, we can derive the 11tlmber of cycli- cally inequivalent Sidel'nikov sequences ns in the following theorem.

Theorem 2: The number of cyclically inequivalent Side15- nikov sequences of period p" - 1 with a fixed value af k~ i s $(p" - 1 ) / ! I .

Prnqfl Let I?" - 1 = LM. Assume that two SideImnikov sequences s(t) and s(' ' { t ) are cyclically erluivalent, i t . . there exists son~e T such that .s(r + r) = s" YO for all t . Then we can say that

for all the elements IY' but oT = - 1 'and cyf"' = - I , In other words, all the fieId elenzents .r pnwibly except for -I and -trfT satisfy

since if x = rv'. N'" E S I . and N" E S c k , then the LHS of (5). becomes w r i k and so does the RHS of (5 ) . If L = 1, clearly ( 5 ) cannot be satisfied by all .r in Fpl b ~ l - I and -oLT unless the two side? are identical as polynomials. For I, > 1, citlce the degree of t l ~ e polynomial (.raT + t)'." - t.lX + 1 )I- i s less than p" -2, the polynomial shouId educe to zero in order to accommodate p" - 2 roorc. The only possible way lo make ( 5 ) an identity is the case when cvT = 1 and r = p' for some 1 .

In fact, when cyr = 1 and c = (Y' E S I inlplies t h a ~

base s "Urn*..

fl'-"dy from Definition 1. two M-ary Sidel'nikov sequences d on different primitive clcrnentq can be distinct. Let and ~ ' ~ ) ( t ) denote the disjoint subcetr in 12) and the se-

LILI~CC in 13). respectively. when the primitive element n is replaced by another primitive element = ~ r " . Clearly. SF' = S d . The relation between s(r ) :~nd s''.'(t) is derived in

fotIowinp lemma.

Thus from (3). we have .s(t) = sl~'"'(f). Therefore the number of cyclically distinct Sidel'nikov sequences with p e n I;o is q!4pi1 - l ) /n .

Noting that Lemma 2 tells us that sip)(r) = ~ - ' , c ( ~ r ) , we may say that the Sidel'nikov sequence has r l ~ so-called 'constant-on-the-cod property when p - I (mndMI, be- cairse st/) = s tp l ) .

4. Counting the Distinct Ar~tocorrelation Distributions of Sidel'nikov Sequences

In this section. the number of dictinct autocorrelation distri- butions of M-ary Sidel'nikov sequences s"'(t). gcdlc, p" -

IEiCE TRANS. FTINDJIMEWALS. VO[..E42-A, NO.] 1.4NL4RY am9

I ) = I . will he derived. From Lemma 2, the a~~tncorrelatiot~ function I?, (T) of

S'" ' ( I ) can he easily obtained from the autocarrelation func- tion R(T) of str) in (3). which is derived in 131.

Thearem 3: The nontrivial (that is, T $ 0 mod y" - I ) au- tocorrelation function R,.(T) or . ~ ~ ' ' ( t ) is given as

where F, denotes the col

Using he result in 131. it can be shou be qiniplitied as

In that R, .(TI can

when V I , ~ ~ ( - I ) = 1 or equivalently (pn - I ) /M is even, and

when ~//~,(- l) = -1 or equivalentIy (p" - 1 )/M is odd. The autocomelation di~tsibutions of s""(t) can he de-

rived as in the folIowin_~ theorem, which can be viewed as the generaIization of the autocorrelation distributions of s(tl derived in 131.

m 4: Let N,(a) be the nuniber of T(# El) such that I , ( , r , - n. Then the out-of-phase autocorrelation distribu- tions of AY-ary Sidel'nikov sequences s" '( t) of period p" - I are given as:

Case 1) i k ~ ( - I ) = 1; 1 ) N , (0) = c:!,' ( (ci . ci + c k i l ) , ~ + (ci, c k ~ ) ~ }

-k (0, C ~ ) M

2) Nc(I?jsi) = (2ci, ci + ~ k n ) , ~ , 1 5 i 5 M - I 3) N,(R,,,) = (ci + c j , ci + rkoIM

-t (ci t r j . c j + c k ~ ) ~ , 1 I i < .j 5 M - 1 . Case 2) 1 ) = - 1 ; 1) NJ-2) = x;l;.lS I(? + ci, ci + ckUlM

+ ($ + ci. +- cko)n,} + (0. 6 - + ~ . k n ) ~ 2) RI,(Qi.i) = (2ci, ci + ~ k o ) ~ { ,

0 1 i < M - I a n d i f M / 2 3) hi,(Qi+j)

= ici + cj , ci + c ' l i ~ ) ~ + (ci + c j , c.j + ckg),t,. 0 1 i < j~ M - l . i + M/2and j + M / 2 . i3

Using Theorems I and 4, wc can derive the number of distinct antocorrelation di strihutions of Side1 ' nikov se- quences as in the following theorem.

Theorem 5: Let p be a prime and Mlp" - 1 . Lel9&, be a ontr trivial multiplicarive character of Fpn. The nurnber of the distinct aurocorrelation distributions of M-ary Sidel'nikov sequences . ~ " " ' ( f ) with gcd(c, p" - 1) = I and a fixed Ro is given as:

Case 1) For M = 2. there is a u n i q ~ ~ e autocorrelatinn distribution.

~ a s e 2 3 1 f ~ > 2 a n d ~ l j ~ ~ + I ) f o r s o m e k . 1 ~k < n,

then the autocorrelation diqtrihution of M-ary Sidel'nikov

sequences is unique. Case 3 ) Tf M > 2 and M + ( p L + 11 for any li, I 5 k < n.

then the number of their distinct autocorrelation distribu- tions is less than or equaI to d(M)/k' for ko + 0. MI2 and ir less than or equal to d( M ) / 2 k f for k,, = O or M / 2 , where k' is the smallest inlegel- satisfying - i i).

iilice M = 2, any c relatively prime to p" - I is odd. r nut;. s\''(f) is just the c-decimation of .r(r). Since the sequences decimat~d by any constant relatively prime to rhe period share the same autoco~rclarion distl-ibution, there is a unique autocorrelation distribution.

Case 2 ) In this cave, n !nust be even and Ic always di- vides 1712. Thus $ M ( - 1) = I since (p" - 1 )/M is even. Using Theorems I and 4, we can derive the f ~ ~ l l o w i n ~ :

( 1 ) ko = O

77' + 77, if i = M / 7 Nr(Ri.i) =

otherwise

2(rT2 + 7)- if i + .j = I% N<,(R,i) =

2q2, otherwise

7 7 2 - ( ~ - 3 ) 1 7 - 1, i f i = M / 2 Nc(Ri,i) =

otherwise

I ( 3 ) h ;t 0 and 1i0 # M / 2 -

v' "r q, if i = k", -ko. MI:! NL.(Ri.j) =

otherwise

nnd i # i R o i f i + j f M

Case 3) For cc~unti~ig the number ot d~stinct autocor- I-elation distributions. it is enough to counl the number of distincr values that Nc(/<i , i ) can take for a given R,., as c

-

LETTER. 345

Table 1 Distinct autncorrelatiun distri hut ions o f 5-ary Sjdel'uib ov sequencr

varies under the condition of gcd(c, pn - 1 ) = 1. Note from Theorem 4, that each hr, (R.,;) i s represented in terms of cy- clotnrnic numbers of order M. Since gcd(r,pn - 11) = 1 implies pcd(c, M ) = 1. it is clear that the number of distinct auivcorrelation distributions is upper bounded by &M).

By Etiler's theorem, we have

IF k' is the smallest integer satisfying ~ I j p ' - I ) , then k'l@(iZI) and c., cp, cp< cP' , . - , rpL'-' are all clistinct nod M. From Lemma I , we have

Thus, the number of distinct autocorreIarion distributions is less than or equal to @ ( M ) / k ' .

From I ) of Lemma 1, for any c and r' such that c+ r' = M , we have

Uci, ci + ~ h ) ~ = (M - 2ci, -ci -I- cknl~r = (2rJ i , c'i - ~ ' k ~ ) ~

(c(i -t . I ) , ci + r k 0 1 ~ -t ( ~ ( i I- , j ) , c j + ckoIi&l = CM - c(j + j). -c,j + rko)n4

+ (M - c(i + j ) , -ri + ibk0jM

= [c'(i I- j), c'i - c 'kdM + [ r f ( i + j). c',j - c ' k ~ ) , ~ .

rhus. w l ~ ~ n ko = 0 or M / 2 , we have N,.(R,,i) = V,..(R,,,J) from Theorem 4. From thc fact that cr P c, cp, rp2. . . . , cpF-' 1, we can say that the number of " ;tinct a~rtocorrelation disttibutinns is upper hounded by

M ) / 2 k ' .

!mark 1: From the numerical analysis for various values M's . p's. and n's, it seems thar the upper bound of Case

i I i s achieved except for the case or M = 8, p = 1 (mod 8).

and ko = 0.4, the number of a~itocot-relation distribution.; of which is cqud to I .

In the following exan-tple, we show rhe case when the number of distinct autocorrelation distributions of the M - a ~ y Sidel'nikov sequences varies according to the value of ko.

Example I: Let N = 1 l 3 - I = 1330 and M = 5. Therc are either two or four distinct nurocorrclation distributions of 5-ary Sidel'nikov sequences as Table 1. •

5. Conclusion

In this letter, we enumerated the number of cyclically in- equivalent Sidel'nikov sequences or period p" - I w~th a fixed value of ko as gS(prl - l ) / n . It is also found that if M 5.1 2 2nd M 7 ( p L + I ) for any li, 1 I k < n , tlie number of their diqrinct aurocorrelatiim distributions is less Ihan or equal to # ( M ) Jk' for kn $ 0. M/2 and is less t1ia11 or equal lo $(M) /2k ' for 1% = (1 or M/2, and otherwise. the autocorreIn- tion distribution of M-ary SldeI'nikov sequencer is unique.

References

111 V M Sldel'ntkw "Sonre k-vulued pieudo-random wquenccc and nenrlv rqu~d~\tnnl codes.'' Prohl. Ini' Tmnvn.. vnl 5. no.]. pp 12-16. f 969

171 A. Lempel. M. Cnhn, all(] W Id Eahm:o~, "A tlass of h.ilanccd htnary scqucncci with opt~rnal autocorrelat~on pmpertles" EBE d m 5 InC Theory. ~01.21 no I, pp.lX42, JJII 1877.

131 Y. S. Kim, J -S Chunp. 3 -5 No, find H Chung. "On the uuto~orrela- tlon d~~fr ib~l1oi l4 (11 Sldel 'n~ho~ xequence\." IEEE Trdnc. Inl' Thcory. vn1.5 I . no 0. pp 37131-1307. Sept 2flM

141 R. L~d l and H. N~edcrrelrer, F~nrte F~eld?. vol 20 of Encycloped~~~ nl Mathernaiicb anil 115 Appl r rn l~nn~ Kencilng. Addison-Wesley. MA. 1983.

[S j T Storcr. Cyclolorny and Difference Set%. Lecturcq rn Advatired M;llhernatrc\. Markhatn, Chicago. IL. 1967

Ihj 1, D Baumen, WII. Mills, L I I ~ R L Wartl. "Unifnrnm cycic~loniy." .I. Nrunber Theory. vo1.14. pp 67-82. 1982.