with low correlation zone - Semantic Scholar
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b
= 1.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
The index set I must be of the form I = A [ B [ C where A
= 1
B
=
fg f +1j 2
C
=
f0 j 2
and
z
z
z
z
C0 ;
C1 ;
z
z
+1
+1
2 0g C
2 1g C
:
[18] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications. Rockville, MD: Computer Science Press, 1985, vol. 1. Revised edition: New York: McGraw-Hill, 1994. [19] T. Storer, Cyclotomy and Difference Sets (Lecture Notes in Advanced Mathematics). Chicago, IL: Markham, 1967. [20] Mobile Station–Base Station Compatibility Standard for Dual-Mode Wideband Spread Spectrum Cellular System, TIA-EIA-IS-95, Telecommun. Ind. Assoc. as a North American 1.5 MHz Cellular CDMA Air-Interface Std., July 1993.
Observe that these three sets are pairwise-disjoint. Therefore,
j j = 1+j j+j j=1+( I
B
C
N=4
0 1) +
N=4
=
N=2:
The other cases can be treated similarly.
A Class of Pseudonoise Sequences over GF Correlation Zone
ACKNOWLEDGMENT The authors wish to thank the anonymous referee for careful review of the original manuscript and helpful comments. Double-check of the computer work by S.-E. Park and C.-Y. Yum at Yonsei University and independently by S.-H. Kim at Seoul National University are greatly appreciated. REFERENCES [1] L. D. Baumert, Cyclic Difference Sets (Lecture Notes in Mathematics). Berlin, Germany: Springer-Verlag, 1971. [2] J. F. Dillon, “Multiplicative difference sets via additive characters,” report, preprint, 1998. [3] H. Dobbertin, “Kasami power functions, permutation polynomials and cyclic difference sets,” in Proc. NATO Advanced Study Institute Workshop: Difference Sets, Sequences and their Correlation Properties, Bad Windshiem, Germany, August 3–14, 1998. [4] S. W. Golomb, Shift-Register Sequences. San Francisco, CA: Holden-Day, 1967. revised edition: Laguna Hills, CA: Aegean Park, 1982. [5] J.-H. Kim and H.-Y. Song, “Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation,” J. Commun. and Networks, vol. 1, no. 1, pp. 14–18, Mar. 1999. [6] A. Lempel, M. Cohn, and W. L. Eastman, “A class of binary sequences with optimal autocorrelation properties,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 38–42, Jan. 1977. [7] R. Lidl and H. Niederreiter, “Finite fields,” in Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20. [8] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [9] J.-S. No, “Generalization of GMW sequences and No sequences,” IEEE Trans. Inform. Theory, vol. 42, pp. 260–262, Jan. 1996. [10] J.-S. No, H. Chung, and M.-S. Yun, “Binary pseudorandom sequences of period 2 1 with ideal autocorrelation generated by the polynomial z + (z + 1) ,” IEEE Trans. Inform. Theory, vol. 44, pp. 1278–1282, May 1999. [11] J.-S. No, S. W. Golomb, G. Gong, H.-K. Lee, and P. Gaal, “Binary pseudorandom sequences of period 2 1 with ideal autocorrelation,” IEEE Trans. Inform. Theory, vol. 44, pp. 814–817, Mar. 1998. [12] J.-S. No, H.-K. Lee, H. Chung, H.-Y. Song, and K. Yang, “Trace representation of Legendre sequences of Mersenne prime period,” IEEE Trans. Inform. Theory, vol. 42, pp. 2254–2255, Nov. 1996. [13] J.-S. No, H.-Y. Song, H. Chung, and K. Yang, “Extension of binary sequences with ideal autocorrelation property,” paper, preprint, 1999. [14] D. V. Sarwate, “Comments on ‘A class of balanced binary sequences with optimal autocorrelation properties’,” IEEE Trans. Inform. Theory, vol. IT-24, Jan. 1978. [15] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [16] R. A. Scholtz and L. R. Welch, “GMW sequences,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 548–553, May 1984. [17] V. M. Sidelnikov, “Some k -valued pseudo-random sequences and nearly equividistant codes,” Probl. Pered. Inform., vol. 5, no. 1, pp. 16–22, 1969. English translation in: Probl. Inform. Transm..
0
0
with Low
Xiaohu H. Tang and Pingzhi Z. Fan, Senior Member, IEEE
Abstract—In this correspondence, a new class of pseudonoise sequences over GF ( ), based on Gordon–Mills–Welch (GMW) sequences, is constructed. The sequences have the property that, in a specified zone, the out-of-phase autocorrelation and cross-correlation values are all equal to 1. Such sequences with low correlation zone (LCZ) are suitable for approximately synchronized code-division multiple-access (CDMA) system. Index Terms—ACF, CCF, Gordon–Mills–Welch (GMW) sequence, low correlation zone (LCZ), zero correlation zone (ZCZ).
I. INTRODUCTION The pseudonoise sequences with low out-of-phase autocorrelation and cross-correlation values are required for direct-sequence (DS) code-division multiple-access (CDMA) (DS-CDMA) system to reduce the multiple-access interference (MAI). -sequences, Gold sequences, Kasami sequences, and Gordon–Mills–Welch (GMW) sequences are well known for their good periodic correlation [1]. A survey on the binary pseudonoise sequences was given in [2], and the related -ary sequences were introduced by [3], [4]. Recently, an approximately synchronized (AS) CDMA (AS-CDMA) system was proposed by Suehiro [5], where the synchronization among users can be controlled within permissible time difference. AS-CDMA system without cochannel interference can be realized by using the sequences with zero correlation zone (ZCZ) [6], [7]. On the other hand, AS-CDMA system with low cochannel interference can be realized by using the sequences with low correlation zone (LCZ), as it is the case of [8]. The binary LCZ sequences introduced in [8] is based on GMW sequences. The correlation values of the sequences are almost all equal to 01 except for a few values. In this correspondence, we have extended the sequences from binary to -ary with the same correlation property. It is shown that the binary sequence set in [8] is only a special case of our result. In the following sections, we will first present the main result of this correspondence, then give a proof of the main result, and finally conclude by an illustrative construction example.
M
p
p
Manuscript received July 13, 1999; revised October 16, 2000. This work was supported by the National Science Foundation of China (NSFC) under Grants 69825102 and 69931050. The authors are with the Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, China (e-mail: xhutang@sina. com; [email protected]). Communicated by S. W. Golomb, Associate Editor for Sequences. Publisher Item Identifier S 0018-9448(01)03000-0.
0018–9448/01$10.00 © 2001 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
II. CONSTRUCTION OF p-ARY LCZ SEQUENCES Similar to ZCZ concept proposed in [6], we can define LCZ as follows.
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where gcd(r; pm 0 1) = gcd(s; pm 0 1) = 1, and r 6= spj mod pm 0 1 for j 2 Z (m 0 1). Similarly, one can get two corresponding m-sequences a0 and b0 over GF (pm )
Definition: Given a sequence set A over GF (p), each sequence having length N , let a, b 2 A, a = (a0 ; a1 ; . . . ; aN 01 ), b = (b0 ; b1 ; . . . ; bN 01 ), and be a constant, then the LCZ Lcz is defined as
Lcz = max fL j jRa; b ( )j
; where (j j < L and a 6= b) or (0 < j j < L and a = b)g
where
Ra; b ( ) =
0
N 1 i=0
a0 = fai0 g = b0 = fb0 g = i
m ri Tr1 ( ) m si Tr1 ( )
which will be used to determine the correlation spectrum of the new sequences. 3) From sequences a and b, we can obtain the following p-ary sequence set:
A = a; a 0 S T b; . . . ; a 0 S (p 02)T b; a 0 S (p 01)T b
a^i+ ^b3i
where S j denotes left-shift operator and S j b denotes j -shift version of sequence b.
and 3 denotes complex conjugation and
We will prove in the next section the following main result.
a^i = exp j 2 ai p
and ^bi = exp
j 2 bi p
and the subscript addition i + is performed modulo N . Normally, it is required that the constant should be as low as possible, for example, = 0, or 1, or 2, etc. When = 0, the LCZ Lcz becomes LCZ Zcz . Moreover, a sequence with LCZ property is called LCZ sequence, a sequence set A is called LCZ sequences set if all the sequences are LCZ sequences. In this correspondence, a new class of pseudonoise LCZ sequences over GF (p) with = 01 will be presented. Before the construction is given, some notations need to be explained. In order to describe the new construction, we need a p-ary GMW sequence defined by
ai = Trm 1
n i r Trm ( )
Where is a primitive root of the finite field GF (p ), and r is an n integer with gcd(r; pm 0 1) = 1. Trm (x) is the trace function, and n with n divisible by m, maps GF (p ) into subfield GF (pm ) according to the relation
0
n=m 1 i=0
T
=
and
= : T
Construction I: 2) The first step of our method is to construct two GMW sequences
a = fai g = b = fbi g =
m
Tr1
n i Trm ( )
N;
01;
pn0m 0 1 n0m Ru; v ( =T ); +p Rx; y ( ) =
01; pn02m (1 + Rb ; a (l1 )) (1 + Rb ; a (l2 )) 0 1;
if = 0 and x = y if = 0 and x 6= y if 6= 0 and = 0 mod T if 6= 0 mod T and (x = a or y = a) if = 6 0 mod T;
x = a 0 Sl T b and y = a 0 S l T b
0 u = a0 ; l 0 a 0S b ; 0 v = a0 ; l 0 a 0S b ;
if x = a if x = a 0 S lT b
if y = a if y = a 0 S lT b:
Since the periodic correlation property of the set A is determined by the related p-ary m-sequences a0 and b0 , by limiting the range of the left-shift operator, we can obtain the following LCZ sequences.
xp :
Let f (x) be a primitive polynomial of with degree n, Z (n) denote the set f0; 1; . . . ; n 0 1g, and
pn 0 1 pm 0 1
2 A, then their periodic correlation function
where the sequences u and v are defined as
: n
n Trm (x) =
Theorem 1: Let x; y Rx; y ( ) is given by
r
s m n i Tr1 Trm ( )
Construction II: 2) This step is identical to Construction I.
3) Compute Rb ; a , suppose there exist M 0 1 time shifts, such that
Rb ; a (l1 ) = Rb ; a (l2 ) = 1 1 1 = Rb ; a (lM 01 ) = 01 then one can obtain a class of p-ary sequences
A = a; a 0 S l T b; a 0 S l T b; . . . ; a 0 S l
T
b :
By Theorem 1, which is our main result, we have the following corollary.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
Corollary 1: Let x; is given by
Rx; y ( ) =
y 2 A; then their periodic correlation function
N; pn0m 0 1 n0m Ru; v ( =T ); +p 01;
if = 0 and x = y if 6= 0 and = 0 mod T else.
Obviously, A is an LCZ sequence set with length N = pn 0 1, low correlation value = 01, and LCZ
Lcz = T
n = (p
x = a and y = a 0 S lT b; 3) x = a 0 S lT b and y = a; 4) x = a 0 S l T b and y = a 0 S l T b; where l1 , l2 2 Z (pm 0 1). 2)
Here, we only consider case 3), the rest are similar. The periodic correlation function between sequence x and y is
Rx; y ( ) =
n k if Trm ( ) = 0 n if Trm ( k ) = e
the sequence e = fe0 ; e1 ; . . . ; ep 02 g is called the shift sequence associated with the primitive polynomial f (x). The following lemma is quoted from [11, Theorem 19]. Lemma: Let P ((a; b); k) be the number of i’s satisfying bi = n i n i+k m Trm (a ) = a, bi+k = Trm (a ) = b, a; b 2 GF (p ), and n 0 i p 0 2. Then we have
pn02m ; pn02m 0 1; P ((a; b); k) =
pn0m ; pn0m 0 1;
if k 6= 0 mod T and (a; b) 6= (0; 0) if k 6= 0 mod T and (a; b) = (0; 0) if k 0 mod T and (a; b) = (a; k a); if k 0 mod T and (a; b) = (0; 0).
i=0
a 6= 0
Corollary 2: Let e = fe0 ; e1 ; . . . ; ep 02 g be the shift sequence associated with the primitive polynomial f (x). For fixed k 2 Z (pn 0 1), k 6= 0 mod T , the list of difference
0 ej (mod pm 0 1): j 2 Z (T ))
contains each element of Z (pm 0 1) exactly pn02m times. Based on the above corollary, we now prove the main theorem: Proof: Let x;
y 2 A. There are three cases to consider:
j 2 Trm p 1
n i+ r Trm ( )
0 Trmn i
+l
s
T +
0 Trmn ( i )r
n i+l T s + Trm ( )
0
T 1 p
02
=
1 exp j 2p Trm ri Trmn ( i )r 0 s i l Trmn ( i )s 0 ri +
1
(
+
s(i
+
)
+l )
+
n i r Trm ( )
n i s Trm ( )
(1)
with i = i1 + i2 T , i1 2 Z (T ) and i2 2 Z (pm 0 1). n k 1) = 0 mod T : By trace function property, Trm (a ) = 0 ocn0m m curs exactly (p 0 1)=(p 0 1) times when k ranges over Z (T ). So, (1) can be simplified as the following form:
Rx; y ( ) = pn0m 0 1 + pn0m
1 exp
p
02
i =0
j 2 Trm r(i p 1
+d+ =T )
0 r i (
n0m =p
0 s i
+d)
(
+
+l +d+ =T )
s(i +l +d)
0 1 + pn0m Ru; v ( =T ) n i d m where Trm ( ) = , d 2 Z (p 0 1). Since
The following corollary can be easily derived from the lemma.
(ej +k
exp
i =0 i =0
In this section, we will give a proof of the main theorem presented in the last section. The well-known GMW sequence was presented in 1962 by three scholars [9]. Later, the cross-correlation function of GMW sequences was analyzed by Games [10], and the cross-correlation function of the corresponding p-ary GMW sequences were investigated in [11]. For k 2 Z (pn 0 1), define
e;
x^i+ y^i3
02
=
III. PROOF OF THE MAIN RESULT
1;
02
i=0 p
0 1)=(pm 0 1):
As for set size M , it can be shown in the next section that, for two special cases, we have M = pm 0 pm0f and M = (pm 0 pm=2 )=2; f will be explained later. It should be noted that the spreading codes presented in [8] is only a special case for p = 2.
ek =
p
Ru; v (0) = Rb ; b (l2 0 l1 ) = 01;
when l1 = 6 l2
we get Rx; y (0) = 01 when x 6= y . 2) 6= 0 mod T : Through the Lemma, for fixed , we can get n n A = i1 jTrm i = 0 and Trm i + = 0; i1 2 Z (T ) kAk = pn02m 0 1 n n i + B = i1 jTrm i = 0 and Trm ( ) 6= 0; i1 2 Z (T ) n0m n02m kBk = p pm00p 1 n n i + i 6= 0 and Trm ( ) = 0; i1 2 Z (T ) C = i1 jTrm n0m n02m kCk = p pm00p 1 n n i Trm i + 6= 0; i1 2 Z (T ) D = i1 jTrm kDk = pn02m (pm 0 1):
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
The cross correlation is given by
Rx; y ( ) =
02
0
T 1 p
i =0 i =0
02
0
T 1p
Let
R(i1 ; i2 jA) +
+ i =0 i =0
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02
0
T 1 p
i =0 i =0
R(i1 ; i2 jC ) +
0
T 1 p
R(i1 ; i2 jB )
02
i =0 i =0
R(i1 ; i2 jD)
n i Trm ( ) =
R(i1 ; i2 ) = exp
RD def =
i +
0 s i l Trmn i r 0 ri Trmn i (
+
RA def = RB def = =
02
0
T 1 p
i =0 i =0
02
0
T 1 p
i =0 i =0
+
s(i
+l )
p
02 exp
i =0
=
0
T 1 p
02
i =0 i =0
p
p
02 exp
i =0
0 s i
w=0
i =0
+d +w )
0 r i n02m =p
p
n02m =p
p
02 p 02
w=0
02
i =0
u^(w)
0 s i
+d )
(
+
+l +d +w )
s(i +l +d )
u^(i2 + w)^v3 (i2 ) p
02
v^3 (w)
r )
u=
m ri Tr1 (
0 s l
+i)
v=
m ri Tr1 (
0 s l
+i)
and
n i + s Trm ( ) +d)
(
(
)
)
:
Hence,
Rx; y ( ) = RA + RB + RC + RD n02m =p 0 1 + pn02m Rb ; a (l1 )
+l +d)
n02m +p Rb ; a (l2 )
n02m +p Rb ; a (l1 )Rb ; a (l2 )
n02m =p (1 + Rb ; a (l1 )) (1 + Rb ; a (l2 ))
0 1:
According to the above induction, we complete the proof. Q.E.D. The size of the LCZ sequence set is determined by the spectrum of the cross correlation between a0 and b0 . So far, we only know the two cases [12], [13]:
i =0
g=
n i r Trm ( )
s(i
j 2 Trm p 1
+l )
0 Rb ; a (l2 ) =p
d , d 2 Z (pm 0 1).
(
+d)
+
pk + 1; 2k (p + 1)=2; 2k p 0 pk + 1;
if p even if p odd or for any p
with g 6= pj mod pm 0 1. Let s = grpl mod pm 0 1, then the cross correlation take the following values:
n i s Trm ( )
0 r i
n 2m
n i where Trm ( ) =
02 p 02
Theorem 2 (Trachtenberg): Let f = gcd(m; k) with m=f odd and
1 exp j 2p Trm 0 ri n02m =p
+
02
+
p
(
2 Z (pm 0 1), and Rb ; a () is real-
1
d :
R(i1 ; i2 jD)
j 2 Trm r(i p 1
s
R(i1 ; i2 jC )
pn0m 0 pn02m pm 0 1
)=
where
s(i +l )
(
n i + d where Trm ( ) = , d valued (see [11, Theorem 1]).
+
w=0 w=0 n02m Rb ; a (l1 )Rb ; a (l2 ) =p
j 2 Trm r(i p 1
n02m 3 =p Rb ; a (l1 ) n02m Rb ; a (l1 ) =p
i =0 i =0
1 exp
R(i1 ; i2 jB )
pn0m 0 pn02m p 02 pm 0 1 i =0 j 2 1 exp p Tr1m ri Trmn ( i
n02m =p
i
n Trm
+
s
R(i1 ; i2 jA) = pn02m 0 1
0
RC def =
)
02
0
T 1 p
n02m =p
r
n Trm
n i Trm (
and
For i1 2 Z (T ), applying Corollary 2, w = d2 0 d1 contains each element of Z (pm 0 1) exactly pn02m times. So, we have
where
j 2 Trm ri p 1
d
s(i +l +d)
01 + p m 01 01 0 p m (
+f )=2
(
+f )=2
occurs (pm0f + p(m0f )=2 )=2 times, occurs (pm 0 pm0f 0 1) times, occurs (pm0f 0 p(m0f )=2 )=2 times.
Applying the result to Corollary 1, we have size M = pm 0 pm0f .
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
TABLE I
LCZ SEQUENCES OF LENGTH
N = 80, SIZE M = 3, AND LCZ L
=
Theorem 3 (Helleseth): Let p be an odd prime, m be even, pm=2 6 , and g pm=2 0 . Let s grpl pm 0 , then the cross correlation take the following values:
2 mod 3
=2
1
01 0 p 01 01 + p 01 + 2p
=
mod
( (
)3 2) 2
(
)6
1
occurs pm 0 pm=2 = times, occurs pm 0 pm=2 0 = times, occurs pm=2 times, occurs pm 0 pm=2 = times.
m=2
m=2 m=2
m
= mod (3 )
1
m
pm 0 for 6 spj GF 4 which satisfies a4 sequences
2 Z (m 0 1), and a primitive root a of + a2 + 2 = 0, we can obtain two GMW
j
a = fa g = Tr12 Tr24 ( ) = f1; 0; 0; 0; 1; 0; 0; 2; 1; 0; 1; 1; 1; 2; 0; 0; 2; 2; 0; 1; 0; 2; 2; 1; 1; 0; 1; 0; 1; 2; 1; 2; 2; 1; 2; 0; 1; 2; 2; 2; 2; 0; 0; 0; 2; 0; 0; 1; 2; 0; 2; 2; 2; 1; 0; 0; 1; 1; 0; 2; 0; 1; 1; 2; 2; 0; 2; 0; 2; 1; 2; 1; 1; 2; 1; 0; 2; 1; 1; 1g i
i
= (p 0 p 2 )=2: m
= 1 and s = 7
gcd (r; p 0 1) = gcd(s; p 0 1) = 1 r
Applying the result to Corollary 1, we have size
M
By choosing r
=9
m=
According to [14], the size M of the LCZ sequence set A satisfying
b = fb g =
MLcz N +1
=
1
1
where N pn 0 is the length of the sequence. On the basis of the discussion, when the size M have
MLcz N +1
=
(p 0 p m
0f ) p 01 p 01
m
p
f
= p 0p m
m
01 , we
0 1 < 1:
p When p ! 1, the ratio MLcz =N + 1 ! 1, sequences set tends to p
n
i
and two corresponding m-sequences
f
be optimal. By Theorem 2, we have
MLcz =N + 1 Moreover, when m Theorem 2, so that
50%; 75%;
a0 = b0 =
Tr12( 10 ) = f2; 1; 0; 1; 1; 2; 0; 2g Tr12( 70 ) = f2; 2; 0; 2; 1; 1; 0; 1g
R
( ) = f2 04 01 04
i
i
with
when m is odd when m
2(mod4).
= 3k(k = 2; 3; . . .), we even take f = k in
b ;a
in this case, MLcz=N
k
k
+ 1 ! 1 with m ! 1.
Rb ; a
k
= 3, m = 2, n = 4, we have T
= pp 00 11 = 10: n
m
;
;
;
5; 2;
01 2g ;
:
(2) =
Rb ; a
(6) =
01
we have l1 = 2, l2 = 6. Therefore, we obtain an LCZ sequence set with N = 34 0 1 = 80, Lcz = T = 10, and M = (33 0 32 )=2 = 3, i.e.,
IV. AN ILLUSTRATIVE EXAMPLE Let p
;
Because there are two 01’s in the spectrum, i.e.,
1 MLcz =N + 1 (p3 0 p2 )=p3 = 1 0 1=p k
7
Tr12 Tr24( ) = f1; 0; 0; 0; 2; 0; 0; 2; 2; 0; 2; 2; 2; 1; 0; 0; 1; 1; 0; 2; 0; 2; 2; 1; 2; 0; 1; 0; 2; 2; 2; 1; 1; 2; 2; 0; 2; 1; 2; 1; 2; 0; 0; 0; 1; 0; 0; 1; 1; 0; 1; 1; 1; 2; 0; 0; 2; 2; 0; 1; 0; 1; 1; 2; 1; 0; 2; 0; 1; 1; 1; 2; 2; 1; 1; 0; 1; 2; 1; 2g i
A
=
f
a; a
0
S
l T
b; a
0
S
l T
g=f
b
a; a
0
S
20
b; a
0
S
60
g
b
as shown in Table I. The ACFs and CCFs of these LCZ sequences are shown in Tables II and III.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
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TABLE II ACFS OF THE LCZ SEQUENCES
TABLE III CCFS OF THE LCZ SEQUENCES
TABLE IV LIST OF PARAMETERS OF THE LCZ SEQUENCES
Finally, a list of parameters of the LCZ sequences, i.e., p; m; n; length N; low correlation zone Lcz ; and size M , is shown in Table IV. REFERENCES [1] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications. New York: Wiley, 1996. [2] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudonoise and realted sequences,” Proc. IEEE, vol. 68, pp. 593–619, 1980. [3] S. C. Liu and J. Komo, “Nonbinary Kasami sequences over GF (p),” IEEE Trans. Inform. Theory, vol. 38, pp. 1409–1412, July 1992. [4] M. Antweiler and L. Bomer, “Complex sequences over GF (p ) with two level autocorrelation function and large linear span,” IEEE Trans. Inform. Theory, vol. 38, pp. 120–130, Jan. 1992. [5] N. Suehiro, “Approximately synchronized CDMA system without cochannel using pseudo-periodic sequences,” in Proc. Int. Symp. Personal Communications’93, Nanjing, China, July 1994, pp. 179–184. [6] P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “A class of binary sequences with zero correlation zone,” Electron. Lett., vol. 35, pp. 777–779, 1999. [7] P. Z. Fan, N. Suehiro, and N. Kuroyanagi, “A novel interference-free CDMA system,” in PIMRC’99, Osaka, Japan, 1999. [8] B. Long, P. Zhang, and J. Hu, “A generalized QS-CDMA system and the design of new spreading codes,” IEEE Trans. Veh. Technol., vol. 47, pp. 1268–1275, Nov. 1998. [9] B. Gordon, W. H. Mills, and L. R. Welch, “Some new different sets,” Can. J. Math., vol. 14, pp. 614–625, 1962. [10] R. A. Games, “Crosscorrelation of m-sequences and GMW sequences with the same primitive polynomial,” Discr. Appl. Math., vol. 16, pp. 139–146, 1985.
[11] M. Antweiler, “Cross-correlation of p-ary GMW sequences,” IEEE Trans. Inform. Theory, vol. 40, pp. 1253–1261, July 1994. [12] H. M. Trachtenberg, “On the cross-correlation function of maximal linear recurring sequences,” Ph.D. disseration, Dep. Elec. Eng., Univ. Southern California, Los Angeles, Jan. 1970. [13] T. Helleseth, “Some results about the cross-correlation function between two maximal linear sequences,” Discr. Math., vol. 16, pp. 209–232, 1976. [14] X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, pp. 551–552, Mar. 2000.
b
= 1.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
The index set I must be of the form I = A [ B [ C where A
= 1
B
=
fg f +1j 2
C
=
f0 j 2
and
z
z
z
z
C0 ;
C1 ;
z
z
+1
+1
2 0g C
2 1g C
:
[18] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications. Rockville, MD: Computer Science Press, 1985, vol. 1. Revised edition: New York: McGraw-Hill, 1994. [19] T. Storer, Cyclotomy and Difference Sets (Lecture Notes in Advanced Mathematics). Chicago, IL: Markham, 1967. [20] Mobile Station–Base Station Compatibility Standard for Dual-Mode Wideband Spread Spectrum Cellular System, TIA-EIA-IS-95, Telecommun. Ind. Assoc. as a North American 1.5 MHz Cellular CDMA Air-Interface Std., July 1993.
Observe that these three sets are pairwise-disjoint. Therefore,
j j = 1+j j+j j=1+( I
B
C
N=4
0 1) +
N=4
=
N=2:
The other cases can be treated similarly.
A Class of Pseudonoise Sequences over GF Correlation Zone
ACKNOWLEDGMENT The authors wish to thank the anonymous referee for careful review of the original manuscript and helpful comments. Double-check of the computer work by S.-E. Park and C.-Y. Yum at Yonsei University and independently by S.-H. Kim at Seoul National University are greatly appreciated. REFERENCES [1] L. D. Baumert, Cyclic Difference Sets (Lecture Notes in Mathematics). Berlin, Germany: Springer-Verlag, 1971. [2] J. F. Dillon, “Multiplicative difference sets via additive characters,” report, preprint, 1998. [3] H. Dobbertin, “Kasami power functions, permutation polynomials and cyclic difference sets,” in Proc. NATO Advanced Study Institute Workshop: Difference Sets, Sequences and their Correlation Properties, Bad Windshiem, Germany, August 3–14, 1998. [4] S. W. Golomb, Shift-Register Sequences. San Francisco, CA: Holden-Day, 1967. revised edition: Laguna Hills, CA: Aegean Park, 1982. [5] J.-H. Kim and H.-Y. Song, “Existence of cyclic Hadamard difference sets and its relation to binary sequences with ideal autocorrelation,” J. Commun. and Networks, vol. 1, no. 1, pp. 14–18, Mar. 1999. [6] A. Lempel, M. Cohn, and W. L. Eastman, “A class of binary sequences with optimal autocorrelation properties,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 38–42, Jan. 1977. [7] R. Lidl and H. Niederreiter, “Finite fields,” in Encyclopedia of Mathematics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20. [8] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [9] J.-S. No, “Generalization of GMW sequences and No sequences,” IEEE Trans. Inform. Theory, vol. 42, pp. 260–262, Jan. 1996. [10] J.-S. No, H. Chung, and M.-S. Yun, “Binary pseudorandom sequences of period 2 1 with ideal autocorrelation generated by the polynomial z + (z + 1) ,” IEEE Trans. Inform. Theory, vol. 44, pp. 1278–1282, May 1999. [11] J.-S. No, S. W. Golomb, G. Gong, H.-K. Lee, and P. Gaal, “Binary pseudorandom sequences of period 2 1 with ideal autocorrelation,” IEEE Trans. Inform. Theory, vol. 44, pp. 814–817, Mar. 1998. [12] J.-S. No, H.-K. Lee, H. Chung, H.-Y. Song, and K. Yang, “Trace representation of Legendre sequences of Mersenne prime period,” IEEE Trans. Inform. Theory, vol. 42, pp. 2254–2255, Nov. 1996. [13] J.-S. No, H.-Y. Song, H. Chung, and K. Yang, “Extension of binary sequences with ideal autocorrelation property,” paper, preprint, 1999. [14] D. V. Sarwate, “Comments on ‘A class of balanced binary sequences with optimal autocorrelation properties’,” IEEE Trans. Inform. Theory, vol. IT-24, Jan. 1978. [15] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [16] R. A. Scholtz and L. R. Welch, “GMW sequences,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 548–553, May 1984. [17] V. M. Sidelnikov, “Some k -valued pseudo-random sequences and nearly equividistant codes,” Probl. Pered. Inform., vol. 5, no. 1, pp. 16–22, 1969. English translation in: Probl. Inform. Transm..
0
0
with Low
Xiaohu H. Tang and Pingzhi Z. Fan, Senior Member, IEEE
Abstract—In this correspondence, a new class of pseudonoise sequences over GF ( ), based on Gordon–Mills–Welch (GMW) sequences, is constructed. The sequences have the property that, in a specified zone, the out-of-phase autocorrelation and cross-correlation values are all equal to 1. Such sequences with low correlation zone (LCZ) are suitable for approximately synchronized code-division multiple-access (CDMA) system. Index Terms—ACF, CCF, Gordon–Mills–Welch (GMW) sequence, low correlation zone (LCZ), zero correlation zone (ZCZ).
I. INTRODUCTION The pseudonoise sequences with low out-of-phase autocorrelation and cross-correlation values are required for direct-sequence (DS) code-division multiple-access (CDMA) (DS-CDMA) system to reduce the multiple-access interference (MAI). -sequences, Gold sequences, Kasami sequences, and Gordon–Mills–Welch (GMW) sequences are well known for their good periodic correlation [1]. A survey on the binary pseudonoise sequences was given in [2], and the related -ary sequences were introduced by [3], [4]. Recently, an approximately synchronized (AS) CDMA (AS-CDMA) system was proposed by Suehiro [5], where the synchronization among users can be controlled within permissible time difference. AS-CDMA system without cochannel interference can be realized by using the sequences with zero correlation zone (ZCZ) [6], [7]. On the other hand, AS-CDMA system with low cochannel interference can be realized by using the sequences with low correlation zone (LCZ), as it is the case of [8]. The binary LCZ sequences introduced in [8] is based on GMW sequences. The correlation values of the sequences are almost all equal to 01 except for a few values. In this correspondence, we have extended the sequences from binary to -ary with the same correlation property. It is shown that the binary sequence set in [8] is only a special case of our result. In the following sections, we will first present the main result of this correspondence, then give a proof of the main result, and finally conclude by an illustrative construction example.
M
p
p
Manuscript received July 13, 1999; revised October 16, 2000. This work was supported by the National Science Foundation of China (NSFC) under Grants 69825102 and 69931050. The authors are with the Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, China (e-mail: xhutang@sina. com; [email protected]). Communicated by S. W. Golomb, Associate Editor for Sequences. Publisher Item Identifier S 0018-9448(01)03000-0.
0018–9448/01$10.00 © 2001 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
II. CONSTRUCTION OF p-ARY LCZ SEQUENCES Similar to ZCZ concept proposed in [6], we can define LCZ as follows.
1645
where gcd(r; pm 0 1) = gcd(s; pm 0 1) = 1, and r 6= spj mod pm 0 1 for j 2 Z (m 0 1). Similarly, one can get two corresponding m-sequences a0 and b0 over GF (pm )
Definition: Given a sequence set A over GF (p), each sequence having length N , let a, b 2 A, a = (a0 ; a1 ; . . . ; aN 01 ), b = (b0 ; b1 ; . . . ; bN 01 ), and be a constant, then the LCZ Lcz is defined as
Lcz = max fL j jRa; b ( )j
; where (j j < L and a 6= b) or (0 < j j < L and a = b)g
where
Ra; b ( ) =
0
N 1 i=0
a0 = fai0 g = b0 = fb0 g = i
m ri Tr1 ( ) m si Tr1 ( )
which will be used to determine the correlation spectrum of the new sequences. 3) From sequences a and b, we can obtain the following p-ary sequence set:
A = a; a 0 S T b; . . . ; a 0 S (p 02)T b; a 0 S (p 01)T b
a^i+ ^b3i
where S j denotes left-shift operator and S j b denotes j -shift version of sequence b.
and 3 denotes complex conjugation and
We will prove in the next section the following main result.
a^i = exp j 2 ai p
and ^bi = exp
j 2 bi p
and the subscript addition i + is performed modulo N . Normally, it is required that the constant should be as low as possible, for example, = 0, or 1, or 2, etc. When = 0, the LCZ Lcz becomes LCZ Zcz . Moreover, a sequence with LCZ property is called LCZ sequence, a sequence set A is called LCZ sequences set if all the sequences are LCZ sequences. In this correspondence, a new class of pseudonoise LCZ sequences over GF (p) with = 01 will be presented. Before the construction is given, some notations need to be explained. In order to describe the new construction, we need a p-ary GMW sequence defined by
ai = Trm 1
n i r Trm ( )
Where is a primitive root of the finite field GF (p ), and r is an n integer with gcd(r; pm 0 1) = 1. Trm (x) is the trace function, and n with n divisible by m, maps GF (p ) into subfield GF (pm ) according to the relation
0
n=m 1 i=0
T
=
and
= : T
Construction I: 2) The first step of our method is to construct two GMW sequences
a = fai g = b = fbi g =
m
Tr1
n i Trm ( )
N;
01;
pn0m 0 1 n0m Ru; v ( =T ); +p Rx; y ( ) =
01; pn02m (1 + Rb ; a (l1 )) (1 + Rb ; a (l2 )) 0 1;
if = 0 and x = y if = 0 and x 6= y if 6= 0 and = 0 mod T if 6= 0 mod T and (x = a or y = a) if = 6 0 mod T;
x = a 0 Sl T b and y = a 0 S l T b
0 u = a0 ; l 0 a 0S b ; 0 v = a0 ; l 0 a 0S b ;
if x = a if x = a 0 S lT b
if y = a if y = a 0 S lT b:
Since the periodic correlation property of the set A is determined by the related p-ary m-sequences a0 and b0 , by limiting the range of the left-shift operator, we can obtain the following LCZ sequences.
xp :
Let f (x) be a primitive polynomial of with degree n, Z (n) denote the set f0; 1; . . . ; n 0 1g, and
pn 0 1 pm 0 1
2 A, then their periodic correlation function
where the sequences u and v are defined as
: n
n Trm (x) =
Theorem 1: Let x; y Rx; y ( ) is given by
r
s m n i Tr1 Trm ( )
Construction II: 2) This step is identical to Construction I.
3) Compute Rb ; a , suppose there exist M 0 1 time shifts, such that
Rb ; a (l1 ) = Rb ; a (l2 ) = 1 1 1 = Rb ; a (lM 01 ) = 01 then one can obtain a class of p-ary sequences
A = a; a 0 S l T b; a 0 S l T b; . . . ; a 0 S l
T
b :
By Theorem 1, which is our main result, we have the following corollary.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
Corollary 1: Let x; is given by
Rx; y ( ) =
y 2 A; then their periodic correlation function
N; pn0m 0 1 n0m Ru; v ( =T ); +p 01;
if = 0 and x = y if 6= 0 and = 0 mod T else.
Obviously, A is an LCZ sequence set with length N = pn 0 1, low correlation value = 01, and LCZ
Lcz = T
n = (p
x = a and y = a 0 S lT b; 3) x = a 0 S lT b and y = a; 4) x = a 0 S l T b and y = a 0 S l T b; where l1 , l2 2 Z (pm 0 1). 2)
Here, we only consider case 3), the rest are similar. The periodic correlation function between sequence x and y is
Rx; y ( ) =
n k if Trm ( ) = 0 n if Trm ( k ) = e
the sequence e = fe0 ; e1 ; . . . ; ep 02 g is called the shift sequence associated with the primitive polynomial f (x). The following lemma is quoted from [11, Theorem 19]. Lemma: Let P ((a; b); k) be the number of i’s satisfying bi = n i n i+k m Trm (a ) = a, bi+k = Trm (a ) = b, a; b 2 GF (p ), and n 0 i p 0 2. Then we have
pn02m ; pn02m 0 1; P ((a; b); k) =
pn0m ; pn0m 0 1;
if k 6= 0 mod T and (a; b) 6= (0; 0) if k 6= 0 mod T and (a; b) = (0; 0) if k 0 mod T and (a; b) = (a; k a); if k 0 mod T and (a; b) = (0; 0).
i=0
a 6= 0
Corollary 2: Let e = fe0 ; e1 ; . . . ; ep 02 g be the shift sequence associated with the primitive polynomial f (x). For fixed k 2 Z (pn 0 1), k 6= 0 mod T , the list of difference
0 ej (mod pm 0 1): j 2 Z (T ))
contains each element of Z (pm 0 1) exactly pn02m times. Based on the above corollary, we now prove the main theorem: Proof: Let x;
y 2 A. There are three cases to consider:
j 2 Trm p 1
n i+ r Trm ( )
0 Trmn i
+l
s
T +
0 Trmn ( i )r
n i+l T s + Trm ( )
0
T 1 p
02
=
1 exp j 2p Trm ri Trmn ( i )r 0 s i l Trmn ( i )s 0 ri +
1
(
+
s(i
+
)
+l )
+
n i r Trm ( )
n i s Trm ( )
(1)
with i = i1 + i2 T , i1 2 Z (T ) and i2 2 Z (pm 0 1). n k 1) = 0 mod T : By trace function property, Trm (a ) = 0 ocn0m m curs exactly (p 0 1)=(p 0 1) times when k ranges over Z (T ). So, (1) can be simplified as the following form:
Rx; y ( ) = pn0m 0 1 + pn0m
1 exp
p
02
i =0
j 2 Trm r(i p 1
+d+ =T )
0 r i (
n0m =p
0 s i
+d)
(
+
+l +d+ =T )
s(i +l +d)
0 1 + pn0m Ru; v ( =T ) n i d m where Trm ( ) = , d 2 Z (p 0 1). Since
The following corollary can be easily derived from the lemma.
(ej +k
exp
i =0 i =0
In this section, we will give a proof of the main theorem presented in the last section. The well-known GMW sequence was presented in 1962 by three scholars [9]. Later, the cross-correlation function of GMW sequences was analyzed by Games [10], and the cross-correlation function of the corresponding p-ary GMW sequences were investigated in [11]. For k 2 Z (pn 0 1), define
e;
x^i+ y^i3
02
=
III. PROOF OF THE MAIN RESULT
1;
02
i=0 p
0 1)=(pm 0 1):
As for set size M , it can be shown in the next section that, for two special cases, we have M = pm 0 pm0f and M = (pm 0 pm=2 )=2; f will be explained later. It should be noted that the spreading codes presented in [8] is only a special case for p = 2.
ek =
p
Ru; v (0) = Rb ; b (l2 0 l1 ) = 01;
when l1 = 6 l2
we get Rx; y (0) = 01 when x 6= y . 2) 6= 0 mod T : Through the Lemma, for fixed , we can get n n A = i1 jTrm i = 0 and Trm i + = 0; i1 2 Z (T ) kAk = pn02m 0 1 n n i + B = i1 jTrm i = 0 and Trm ( ) 6= 0; i1 2 Z (T ) n0m n02m kBk = p pm00p 1 n n i + i 6= 0 and Trm ( ) = 0; i1 2 Z (T ) C = i1 jTrm n0m n02m kCk = p pm00p 1 n n i Trm i + 6= 0; i1 2 Z (T ) D = i1 jTrm kDk = pn02m (pm 0 1):
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
The cross correlation is given by
Rx; y ( ) =
02
0
T 1 p
i =0 i =0
02
0
T 1p
Let
R(i1 ; i2 jA) +
+ i =0 i =0
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02
0
T 1 p
i =0 i =0
R(i1 ; i2 jC ) +
0
T 1 p
R(i1 ; i2 jB )
02
i =0 i =0
R(i1 ; i2 jD)
n i Trm ( ) =
R(i1 ; i2 ) = exp
RD def =
i +
0 s i l Trmn i r 0 ri Trmn i (
+
RA def = RB def = =
02
0
T 1 p
i =0 i =0
02
0
T 1 p
i =0 i =0
+
s(i
+l )
p
02 exp
i =0
=
0
T 1 p
02
i =0 i =0
p
p
02 exp
i =0
0 s i
w=0
i =0
+d +w )
0 r i n02m =p
p
n02m =p
p
02 p 02
w=0
02
i =0
u^(w)
0 s i
+d )
(
+
+l +d +w )
s(i +l +d )
u^(i2 + w)^v3 (i2 ) p
02
v^3 (w)
r )
u=
m ri Tr1 (
0 s l
+i)
v=
m ri Tr1 (
0 s l
+i)
and
n i + s Trm ( ) +d)
(
(
)
)
:
Hence,
Rx; y ( ) = RA + RB + RC + RD n02m =p 0 1 + pn02m Rb ; a (l1 )
+l +d)
n02m +p Rb ; a (l2 )
n02m +p Rb ; a (l1 )Rb ; a (l2 )
n02m =p (1 + Rb ; a (l1 )) (1 + Rb ; a (l2 ))
0 1:
According to the above induction, we complete the proof. Q.E.D. The size of the LCZ sequence set is determined by the spectrum of the cross correlation between a0 and b0 . So far, we only know the two cases [12], [13]:
i =0
g=
n i r Trm ( )
s(i
j 2 Trm p 1
+l )
0 Rb ; a (l2 ) =p
d , d 2 Z (pm 0 1).
(
+d)
+
pk + 1; 2k (p + 1)=2; 2k p 0 pk + 1;
if p even if p odd or for any p
with g 6= pj mod pm 0 1. Let s = grpl mod pm 0 1, then the cross correlation take the following values:
n i s Trm ( )
0 r i
n 2m
n i where Trm ( ) =
02 p 02
Theorem 2 (Trachtenberg): Let f = gcd(m; k) with m=f odd and
1 exp j 2p Trm 0 ri n02m =p
+
02
+
p
(
2 Z (pm 0 1), and Rb ; a () is real-
1
d :
R(i1 ; i2 jD)
j 2 Trm r(i p 1
s
R(i1 ; i2 jC )
pn0m 0 pn02m pm 0 1
)=
where
s(i +l )
(
n i + d where Trm ( ) = , d valued (see [11, Theorem 1]).
+
w=0 w=0 n02m Rb ; a (l1 )Rb ; a (l2 ) =p
j 2 Trm r(i p 1
n02m 3 =p Rb ; a (l1 ) n02m Rb ; a (l1 ) =p
i =0 i =0
1 exp
R(i1 ; i2 jB )
pn0m 0 pn02m p 02 pm 0 1 i =0 j 2 1 exp p Tr1m ri Trmn ( i
n02m =p
i
n Trm
+
s
R(i1 ; i2 jA) = pn02m 0 1
0
RC def =
)
02
0
T 1 p
n02m =p
r
n Trm
n i Trm (
and
For i1 2 Z (T ), applying Corollary 2, w = d2 0 d1 contains each element of Z (pm 0 1) exactly pn02m times. So, we have
where
j 2 Trm ri p 1
d
s(i +l +d)
01 + p m 01 01 0 p m (
+f )=2
(
+f )=2
occurs (pm0f + p(m0f )=2 )=2 times, occurs (pm 0 pm0f 0 1) times, occurs (pm0f 0 p(m0f )=2 )=2 times.
Applying the result to Corollary 1, we have size M = pm 0 pm0f .
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001
TABLE I
LCZ SEQUENCES OF LENGTH
N = 80, SIZE M = 3, AND LCZ L
=
Theorem 3 (Helleseth): Let p be an odd prime, m be even, pm=2 6 , and g pm=2 0 . Let s grpl pm 0 , then the cross correlation take the following values:
2 mod 3
=2
1
01 0 p 01 01 + p 01 + 2p
=
mod
( (
)3 2) 2
(
)6
1
occurs pm 0 pm=2 = times, occurs pm 0 pm=2 0 = times, occurs pm=2 times, occurs pm 0 pm=2 = times.
m=2
m=2 m=2
m
= mod (3 )
1
m
pm 0 for 6 spj GF 4 which satisfies a4 sequences
2 Z (m 0 1), and a primitive root a of + a2 + 2 = 0, we can obtain two GMW
j
a = fa g = Tr12 Tr24 ( ) = f1; 0; 0; 0; 1; 0; 0; 2; 1; 0; 1; 1; 1; 2; 0; 0; 2; 2; 0; 1; 0; 2; 2; 1; 1; 0; 1; 0; 1; 2; 1; 2; 2; 1; 2; 0; 1; 2; 2; 2; 2; 0; 0; 0; 2; 0; 0; 1; 2; 0; 2; 2; 2; 1; 0; 0; 1; 1; 0; 2; 0; 1; 1; 2; 2; 0; 2; 0; 2; 1; 2; 1; 1; 2; 1; 0; 2; 1; 1; 1g i
i
= (p 0 p 2 )=2: m
= 1 and s = 7
gcd (r; p 0 1) = gcd(s; p 0 1) = 1 r
Applying the result to Corollary 1, we have size
M
By choosing r
=9
m=
According to [14], the size M of the LCZ sequence set A satisfying
b = fb g =
MLcz N +1
=
1
1
where N pn 0 is the length of the sequence. On the basis of the discussion, when the size M have
MLcz N +1
=
(p 0 p m
0f ) p 01 p 01
m
p
f
= p 0p m
m
01 , we
0 1 < 1:
p When p ! 1, the ratio MLcz =N + 1 ! 1, sequences set tends to p
n
i
and two corresponding m-sequences
f
be optimal. By Theorem 2, we have
MLcz =N + 1 Moreover, when m Theorem 2, so that
50%; 75%;
a0 = b0 =
Tr12( 10 ) = f2; 1; 0; 1; 1; 2; 0; 2g Tr12( 70 ) = f2; 2; 0; 2; 1; 1; 0; 1g
R
( ) = f2 04 01 04
i
i
with
when m is odd when m
2(mod4).
= 3k(k = 2; 3; . . .), we even take f = k in
b ;a
in this case, MLcz=N
k
k
+ 1 ! 1 with m ! 1.
Rb ; a
k
= 3, m = 2, n = 4, we have T
= pp 00 11 = 10: n
m
;
;
;
5; 2;
01 2g ;
:
(2) =
Rb ; a
(6) =
01
we have l1 = 2, l2 = 6. Therefore, we obtain an LCZ sequence set with N = 34 0 1 = 80, Lcz = T = 10, and M = (33 0 32 )=2 = 3, i.e.,
IV. AN ILLUSTRATIVE EXAMPLE Let p
;
Because there are two 01’s in the spectrum, i.e.,
1 MLcz =N + 1 (p3 0 p2 )=p3 = 1 0 1=p k
7
Tr12 Tr24( ) = f1; 0; 0; 0; 2; 0; 0; 2; 2; 0; 2; 2; 2; 1; 0; 0; 1; 1; 0; 2; 0; 2; 2; 1; 2; 0; 1; 0; 2; 2; 2; 1; 1; 2; 2; 0; 2; 1; 2; 1; 2; 0; 0; 0; 1; 0; 0; 1; 1; 0; 1; 1; 1; 2; 0; 0; 2; 2; 0; 1; 0; 1; 1; 2; 1; 0; 2; 0; 1; 1; 1; 2; 2; 1; 1; 0; 1; 2; 1; 2g i
A
=
f
a; a
0
S
l T
b; a
0
S
l T
g=f
b
a; a
0
S
20
b; a
0
S
60
g
b
as shown in Table I. The ACFs and CCFs of these LCZ sequences are shown in Tables II and III.
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TABLE II ACFS OF THE LCZ SEQUENCES
TABLE III CCFS OF THE LCZ SEQUENCES
TABLE IV LIST OF PARAMETERS OF THE LCZ SEQUENCES
Finally, a list of parameters of the LCZ sequences, i.e., p; m; n; length N; low correlation zone Lcz ; and size M , is shown in Table IV. REFERENCES [1] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications. New York: Wiley, 1996. [2] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudonoise and realted sequences,” Proc. IEEE, vol. 68, pp. 593–619, 1980. [3] S. C. Liu and J. Komo, “Nonbinary Kasami sequences over GF (p),” IEEE Trans. Inform. Theory, vol. 38, pp. 1409–1412, July 1992. [4] M. Antweiler and L. Bomer, “Complex sequences over GF (p ) with two level autocorrelation function and large linear span,” IEEE Trans. Inform. Theory, vol. 38, pp. 120–130, Jan. 1992. [5] N. Suehiro, “Approximately synchronized CDMA system without cochannel using pseudo-periodic sequences,” in Proc. Int. Symp. Personal Communications’93, Nanjing, China, July 1994, pp. 179–184. [6] P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “A class of binary sequences with zero correlation zone,” Electron. Lett., vol. 35, pp. 777–779, 1999. [7] P. Z. Fan, N. Suehiro, and N. Kuroyanagi, “A novel interference-free CDMA system,” in PIMRC’99, Osaka, Japan, 1999. [8] B. Long, P. Zhang, and J. Hu, “A generalized QS-CDMA system and the design of new spreading codes,” IEEE Trans. Veh. Technol., vol. 47, pp. 1268–1275, Nov. 1998. [9] B. Gordon, W. H. Mills, and L. R. Welch, “Some new different sets,” Can. J. Math., vol. 14, pp. 614–625, 1962. [10] R. A. Games, “Crosscorrelation of m-sequences and GMW sequences with the same primitive polynomial,” Discr. Appl. Math., vol. 16, pp. 139–146, 1985.
[11] M. Antweiler, “Cross-correlation of p-ary GMW sequences,” IEEE Trans. Inform. Theory, vol. 40, pp. 1253–1261, July 1994. [12] H. M. Trachtenberg, “On the cross-correlation function of maximal linear recurring sequences,” Ph.D. disseration, Dep. Elec. Eng., Univ. Southern California, Los Angeles, Jan. 1970. [13] T. Helleseth, “Some results about the cross-correlation function between two maximal linear sequences,” Discr. Math., vol. 16, pp. 209–232, 1976. [14] X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, pp. 551–552, Mar. 2000.
