A Class of Pseudonoise Sequences over GF Correlation Zone

Transcription

1 1644 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001 b 1. The index set I must be of the form I A [ B [ C where A f1g B fz 1j z 2 C 0; z 12 C 0g and C f0z j z 2 C 1; z 12 C 1g: Observe that these three sets are pairwise-disjoint. Therefore, [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, [19] T. Storer, Cyclotomy and Difference Sets (Lecture Notes in Advanced Mathematics). Chicago, IL: Markham, [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 jij 1jBj jcj 1(N4 0 1) N4 N2: The other cases can be treated similarly. 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, [2] J. F. Dillon, Multiplicative difference sets via additive characters, report, preprint, [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, [4] S. W. Golomb, Shift-Register Sequences. San Francisco, CA: Holden-Day, revised edition: Laguna Hills, CA: Aegean Park, [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 , Mar [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 , Jan [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, [9] J.-S. No, Generalization of GMW sequences and No sequences, IEEE Trans. Inform. Theory, vol. 42, pp , Jan [10] J.-S. No, H. Chung, and M.-S. Yun, Binary pseudorandom sequences of period 2 01 with ideal autocorrelation generated by the polynomial z (z 1), IEEE Trans. Inform. Theory, vol. 44, pp , May [11] J.-S. No, S. W. Golomb, G. Gong, H.-K. Lee, and P. Gaal, Binary pseudorandom sequences of period 2 01 with ideal autocorrelation, IEEE Trans. Inform. Theory, vol. 44, pp , Mar [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 , Nov [13] J.-S. No, H.-Y. Song, H. Chung, and K. Yang, Extension of binary sequences with ideal autocorrelation property, paper, preprint, [14] D. V. Sarwate, Comments on A class of balanced binary sequences with optimal autocorrelation properties, IEEE Trans. Inform. Theory, vol. IT-24, Jan [15] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, vol. 68, pp , May [16] R. A. Scholtz and L. R. Welch, GMW sequences, IEEE Trans. Inform. Theory, vol. IT-30, pp , May [17] V. M. Sidelnikov, Some k-valued pseudo-random sequences and nearly equividistant codes, Probl. Pered. Inform., vol. 5, no. 1, pp , English translation in: Probl. Inform. Transm.. A Class of Pseudonoise Sequences over GF Correlation Zone 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). M -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 p-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 p-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. Manuscript received July 13, 1999; revised October 16, This work was supported by the National Science Foundation of China (NSFC) under Grants and The authors are with the Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan , China ( xhutang@sina. com; p.fan@ieee.org). Communicated by S. W. Golomb, Associate Editor for Sequences. Publisher Item Identifier S (01) /01$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY II. CONSTRUCTION OF p-ary LCZ SEQUENCES Similar to ZCZ concept proposed in [6], we can ine LCZ as follows. Definition: Given a sequence set A over GF (p), each sequence having length N, let a, b 2 A, a (a 0 ;a 1 ;...;a N01), b (b 0;b 1;...;b N01), and be a constant, then the LCZ L cz is ined as L cz maxfl jjr a; b ( )j ; where (j j <L and a 6 b) where R a; b ( ) N01 i0 and 3 denotes complex conjugation and ^a i exp or (0 < j j <L and a b)g ^a i^b 3 i j 2 p ai and ^b i exp j 2 p bi where gcd(r; ) gcd(s; ) 1, and r 6 sp j mod for j 2 Z(m 0 1). Similarly, one can get two corresponding m-sequences a 0 and b 0 over GF (p m ) a 0 fa 0 ig Tr m 1 ( ri ) b 0 fb 0 ig Tr m 1 ( si ) 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;...;a0 S ()T b; a 0 S (p 01)T b where S j denotes left-shift operator and S j b denotes j-shift version of sequence b. We will prove in the next section the following main result. Theorem 1: Let x; y 2 A, then their periodic correlation function R x; y ( ) is given by 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,or1, or2, etc. When 0, the LCZ L cz becomes LCZ Z cz. 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 ined by R x; y( ) N; if 0and x y 01; if 0and x 6 y p n0m 0 1 p n0m R u; v (T); if 6 0 and 0 mod T 01; if 6 0 mod T and (x a or y a) p n02m (1 R b ;a (l 1 )) (1 R b ;a (l 2)) 0 1; if 6 0 mod T; x a 0 S l T b and y a 0 S l T b a i Tr m 1 Tr n m( i ) r : Where is a primitive root of the finite field GF (p n ), and r is an integer with gcd(r; ) 1. Tr n m(x) is the trace function, and with n divisible by m, maps GF (p n ) into subfield GF (p m ) according to the relation nm01 Trm(x) n x p : Let f (x) be a primitive polynomial of with degree n, Z(n) denote the set f0; 1;...;n0 1g, and T pn 0 1 i0 and T : where the sequences u and v are ined as u a0 ; if x a a 0 0 S l b 0 ; if x a 0 S lt b v a0 ; if y a a 0 0 S l b 0 ; 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 a 0 and b 0, by limiting the range of the left-shift operator, we can obtain the following LCZ sequences. Construction II: 2) This step is identical to Construction I. 3) Compute R b ;a, suppose there exist M 01 time shifts, such that R b ;a (l 1)R b ;a (l 2)111 R b ;a (l M01) 01 Construction I: 2) The first step of our method is to construct two GMW sequences a fa i g Tr m 1 Trm( n i r ) b fb ig m Tr 1 Trm( n i s ) then one can obtain a class of p-ary sequences A a; a 0 S l T b; a 0 S l T b;...;a0 S l T b : By Theorem 1, which is our main result, we have the following corollary.

3 1646 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 2001 Corollary 1: Let x; y 2 A; then their periodic correlation function is given by R x; y( ) N; if 0and x y p n0m 0 1 p n0m R u; v (T); if 6 0and 0modT 01; else. Obviously, A is an LCZ sequence set with length N p n 0 1,low correlation value 01, and LCZ L cz T (p n 0 1)(): As for set size M, it can be shown in the next section that, for two special cases, we have M p m 0 p m0f and M (p m 0 p m2 )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. III. PROOF OF THE MAIN RESULT 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(p n 0 1), ine e k 1; if Trn m( k )0 e; if Tr n m( k ) e the sequence e fe 0 ;e 1 ;...;e 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 b i Tr n m(a i ) a, b ik Tr n m(a ik ) b, a; b 2 GF (p m ), and 0 i p n 0 2. Then we have P ((a; b); k) p n02m ; p n02m 0 1; p n0m ; p n0m 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 0modT and (a; b) (a; k a); a6 0 if k 0modT and (a; b) (0; 0). The following corollary can be easily derived from the lemma. Corollary 2: Let e fe 0;e 1;...;e g be the shift sequence associated with the primitive polynomial f (x). For fixed k 2 Z(p n 0 1), k 6 0 mod T, the list of difference (e jk 0 e j (mod): j 2 Z(T )) contains each element of Z() exactly p n02m times. Based on the above corollary, we now prove the main theorem: Proof: Let x; y 2 A. There are three cases to consider: 2) 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 l 1, l 2 2 Z(). Here, we only consider case 3), the rest are similar. The periodic correlation function between sequence x and y is R x; y( ) i0 i0 ^x i ^y 3 i exp T 01 p Trm 1 Tr n m( i ) r 0 Tr n m il T s 0 Tr n m( i ) r Tr n m( il T ) s p Trm 1 ri Tr n m( i ) r 0 s(i l ) Tr n m( i ) s 0 ri Tr n m( i ) r s(i l ) Tr n m( i ) s (1) with i i 1 i 2T, i 1 2 Z(T ) and i 2 2 Z(). 1) 0 mod T : By trace function property, Tr n m(a k )0occurs exactly (p n0m 0 1)() times when k ranges over Z(T ). So, (1) can be simplified as the following form: R x; y ( )p n0m 0 1p n0m p Trm 1 r(i dt) s(i l dt) 0 p n0m 0 1p n0m R u; v (T) where Tr n m( i ) d, d 2 Z(). Since 0 r(i d) s(i l d) R u; v (0) R b ;b (l 2 0 l 1 )01; when l 1 6 l 2 we get R x; y (0) 01 when x 6 y. 2) 6 0 mod T : Through the Lemma, for fixed, we can get A i 1jTr n m and Tr n m i 0;i 1 2 Z(T ) kak p n02m 0 1 B i 1jTrm n and Trm( n i ) 6 0;i 1 2 Z(T ) kbk pn0m 0 p n02m C i 1 jtrm n i 60 and Trm( n i )0;i 1 2 Z(T ) kck pn0m 0 p n02m D i 1 jtrm n i Trm n i 60;i 1 2 Z(T ) kdk p n02m ():

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY The cross correlation is given by R x; y( ) where T01 R(i 1 ;i 2 )exp R A R B T01 T01 R(i 1;i 2jA) R(i 1 ;i 2 jc) T01 T01 p Trm 1 ri Tr n m i T01 pn0m 0 p n02m r R(i 1;i 2jB) R(i 1 ;i 2 jd) 0 s(i l ) Tr n m i s 0 ri Tr n m i r s(i l ) Tr n m i s R(i 1 ;i 2 ja) p n02m 0 1 R(i 1;i 2jB) p Trm 1 ri Tr n m( i ) r p n02m exp p n02m Rb 3 ;a (l 1 ) p n02m R b ;a (l 1) 0 s(i l ) Tr n m( i ) s p Trm r(i d) 1 s(i l d) 0 where Tr n m( i ) d, d 2 Z(), and R b ;a () is realvalued (see [11, Theorem 1]). R C T 01 pn0m 0 p n02m R(i 1;i 2jC) p Trm 1 0 ri Tr n m( i ) r s(i l ) Tr n m( i ) s Let Tr n m( i ) d and Tr n m( i ) d : For i 1 2 Z(T ), applying Corollary 2, w d 2 0 d 1 contains each element of Z() exactly p n02m times. So, we have where and R D Hence, T 01 p n02m w0 R(i 1 ;i 2 jd) p Trm 1 r(i d w) s(i l d w) 0 p n02m w0 p n02m w0 ^u(w) 0 r(i d ) s(i l d ) ^u(i 2 w)^v 3 (i 2) w0 p n02m R b ;a (l 1)R b ;a (l 2) ^v 3 (w) u Tr m 1 ( ri 0 s(l i) ) v Tr m 1 ( ri 0 s(l i) ) : R x; y ( )R A R B R C R D p n02m 0 1p n02m R b ;a (l 1) p n02m R b ;a (l 2) p n02m R b ;a (l 1)R b ;a (l 2) p n02m (1 R b ;a (l 1 )) (1 R b ;a (l 2 )) 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 a 0 and b 0. So far, we only know the two cases [12], [13]: Theorem 2 (Trachtenberg): Let f gcd(m; k) with mf odd and g p k 1; if p even (p 2k 1)2; if p odd or p 2k 0 p k 1; for any p with g 6 p j mod. Let s grp l mod, then the cross correlation take the following values: p n02m exp p n02m R b ;a (l 2) p Trm 1 0 r(i d) s(i l d) 01 p (mf)2 occurs (p m0f p (m0f)2 )2 times, 01 occurs (p m 0 p m0f 0 1) times, 01 0 p (mf)2 occurs (p m0f 0 p (m0f)2 )2 times. where Tr n m( i ) d, d 2 Z(). Applying the result to Corollary 1, we have size M p m 0 p m0f.

5 1648 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 9 Theorem 3 (Helleseth): Let p be an odd prime, m be even, p m2 6 2mod3, and g 2p m Let s grp l mod, then the cross correlation take the following values: 01 0 p m2 occurs (p m 0 p m2 )3 times, 01 occurs (p m 0 p m2 0 2)2 times, 01 p m2 occurs p m2 times, 01 2p m2 occurs (p m 0 p m2 )6 times. By choosing r 1and s 7 gcd (r; ) gcd(s; ) 1 r 6 sp j mod for j 2 Z(m 0 1), and a primitive root a of GF (3 4 ) which satisfies a 4 a 2 2 0, we can obtain two GMW sequences Applying the result to Corollary 1, we have size M (p m 0 p m2 )2: According to [14], the size M of the LCZ sequence set A satisfying ML cz N 1 1 where N p n 0 1 is the length of the sequence. On the basis of the discussion, when the size M p m 0 p m01,we have ML cz (p m 0 p m0f ) N 1 p 01 p 01 p n pf 0 1 p f < 1: When p!1, the ratio ML cz N 1! 1, sequences set tends to be optimal. By Theorem 2, we have a fa i g Tr 2 1 Tr2( 4 i ) 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 b fb i g Tr 2 1 Tr 4 2( i ) 7 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 and two corresponding m-sequences a 0 Tr1( 2 10i ) f2; 1; 0; 1; 1; 2; 0; 2g b 0 Tr1( 2 70i ) f2; 2; 0; 2; 1; 1; 0; 1g ML cz N 1 50%; when m is odd 75%; when m 2(mod4). with R b ;a ()f2; 04; 01; 04; 5; 2; 01; 2g: Moreover, when m 3k(k 2; 3;...), we even take f k in Theorem 2, so that 1 ML czn 1 (p 3k 0 p 2k )p 3k 10 1p k in this case, MLczN 1! 1 with m!1. IV. AN ILLUSTRATIVE EXAMPLE Let p 3, m 2, n 4,wehave Because there are two 01 s in the spectrum, i.e., R b ;a (2) R b ;a (6) 01 we have l 1 2, l 2 6. Therefore, we obtain an LCZ sequence set with N , L cz T 10, and M ( )23, i.e., A fa; a 0 S l T b; a 0 S l T bg fa; a 0 S 20 b; a 0 S 60 bg T pn : as shown in Table I. The ACFs and CCFs of these LCZ sequences are shown in Tables II and III.

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 4, MAY 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 L cz ; and size M, is shown in Table IV. REFERENCES [1] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications. New York: Wiley, [2] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudonoise and realted sequences, Proc. IEEE, vol. 68, pp , [3] S. C. Liu and J. Komo, Nonbinary Kasami sequences over GF (p), IEEE Trans. Inform. Theory, vol. 38, pp , July [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 , Jan [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 [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 , [7] P. Z. Fan, N. Suehiro, and N. Kuroyanagi, A novel interference-free CDMA system, in PIMRC 99, Osaka, Japan, [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 , Nov [9] B. Gordon, W. H. Mills, and L. R. Welch, Some new different sets, Can. J. Math., vol. 14, pp , [10] R. A. Games, Crosscorrelation of m-sequences and GMW sequences with the same primitive polynomial, Discr. Appl. Math., vol. 16, pp , [11] M. Antweiler, Cross-correlation of p-ary GMW sequences, IEEE Trans. Inform. Theory, vol. 40, pp , July [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 [13] T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discr. Math., vol. 16, pp , [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 , Mar