On the Cross-Correlation of a p-ary m-sequence of Period p 2m 1 and Its Decimated

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 3, MARCH On the Cross-Correlation of a p-ary m-sequence of Period p m 1 Its Decimated Sequences by (p m +1) =(p +1) Sung-Tai Choi, Taehyung Lim, Jong-Seon No, Fellow, IEEE, Habong Chung, Member, IEEE Abstract In this paper, for an odd prime, we investigate into the cross-correlation of a -ary -sequence ( ) of period 1 +1 its -decimated sequences ( + ), 0, where = ( +1) ( +1), =, is an odd integer There are +1 distinct decimated sequences ( + ) since gcd( 1) = ( +1) It is shown that the magnitude of the cross-correlation values is upper bounded by We also construct the sequence family from these sequences, where the family size is the correlation magnitude is upper bounded by Index Terms Cross-correlation, decimated sequence, -sequence, nonbinary sequence, quadratic form, sequence family I INTRODUCTION T HERE has been much research to find a decimation value such that the cross-correlation of a -ary -sequence its decimated sequence is low The values with have been studied in [1], [], [3], [4] There have also been some research works dealing with a decimation factor not relatively prime to the period When is not relatively prime to the period, the decimated sequences have short period, For a ternary case, in [5], the authors derived the correlation distribution for, which corresponds to the Coulter Matthews decimation In [6], the authors derived the correlation distribution of with, where is odd is an odd prime In [7], the author showed that the magnitude of correlation values is upper bounded by for for ternary -sequences In [8], the authors extended Muller s result to any odd prime case, ie, for Manuscript received April 30, 010; revised May 5, 011; accepted August 30, 011 Date of current version February 9, 01 The material in this paper was presented at the 010 IEEE International Symposium on Information Theory This work was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science Technology, Korea Government under Grant the Korea Communications Commission (KCC), Korea, under the R&D program supervised by the Korea Communications Agency (KCA) under Grant KCA S-T Choi, T Lim, J-S No are with the Department of Electrical Engineering Computer Science, INMC, Seoul National University, Seoul , Korea ( stchoi@cclsnuackr; jayelish@hotmailcom; jsno@snuac kr) H Chung is with the School of Electronics Electrical Engineering, Hong-Ik University, Seoul , Korea ( habchung@hongikackr) Communicated by N Yul Yu, Associate Editor for Sequences Digital Object Identier /TIT derived the upper bound as In [1], the authors derived the correlation distribution for, when is an odd prime The known decimation values the upper bounds of magnitude of cross-correlation values are listed in Table I In this paper, for an odd prime, the upper bound on the magnitude of cross-correlation values of a -ary -sequence its decimation sequences,, is derived for,, an odd integer The decimated sequences have the period of because It is shown that the magnitude of cross-correlation function of is upper bounded by We also construct the sequence family by using, where the family size is the magnitude of correlation values is upper bounded by II PRELIMINARIES Let be an odd prime the finite field with elements Then, the trace function from to is defined as where Let be a primitive element of Then, a -ary -sequence with the period of can be expressed as Since, there are distinct decimated sequences of period,, which are defined as The cross-correlation function between two at sht is defined as where is a primitive th root of unity The quadratic character of is defined as is a nonzero square in, is a nonsquare in, (1) -ary sequences /$ IEEE

2 1874 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 3, MARCH 01 TABLE I PARAMETERS OF SOME KNOWN DECIMATION VALUES FOR THE NONBINARY CASE The Gauss sum pertaining to the quadratic character of is given as the following theorem Theorem 1 (Theorem 515 [15]): Let be an odd prime the quadratic character of Then, where is a primitive th root of unity A quadratic form over in indeterminates is a homogeneous polynomial in of degree can be expressed as Let denote an -dimensional vector space over The number of solutions satisfying the quadratic form for any can be determined from the rank of the quadratic form The following lemma explains how to determine the rank of a quadratic form Lemma (Lemma 6 [7]): Let be a quadratic form Define () Then, is a subspace of the rank of is defined as A quadratic form in can be regarded as a mapping from into Thus, we will also use the term quadratic form" for this mapping in a finite extension field In this case, Lemma in a finite-field version can be restated as follows Corollary 3: The rank of the quadratic form from to can be determined by finding the number of coordinates that the form is independent of, ie, is the number of, such that, for all A quadratic form in indeterminates over is said to be nondegenerate it has a full rank, ie, can be expressed as the canonical form for Let denote the determinant of the quadratic form If is a nondegenerate quadratic form of rank, the number of solutions in satisfying is determined as in the following lemma Lemma 4 (Theorems [15]): Let be the quadratic character of The number of solutions of in, when is a nondegenerate quadratic form in rank with determinant, is given as follows: Case 1) even; Case ) odd; If a nonzero quadratic form has a rank, it can be rewritten as an equivalent canonical form,, where all [15] Hence, for any, the number of solutions of in is times the number of solutions of the same equation in For a quadratic form with rank, we can decide the number of solutions in, where, from Lemma 4 the previous fact Using Theorem 1 Lemma 4, the following corollary can be stated without proof Corollary 5: Let be a mapping from to corresponding to a quadratic form of rank with determinant Then, the exponential sum is given as follows Case 1) even; Case ) odd;

3 CHOI et al: ON THE CROSS-CORRELATION OF A -ARY -SEQUENCE OF PERIOD 1875 Let be a prime power A polynomial of the form Since is an even integer, we have with coefficients in an extension field of is called a -polynomial or linearized polynomial over If is an arbitrary extension field of, which contains all the roots of, then Thus, we have Hence, (5) can be rewritten as Hence, the set of solutions of in is considered as a vector subspace over, ie, the number of solutions is a power of In the remaining part of this paper, the following notations will be used: 1) is an odd integer; ) ; 3) is a primitive element of ; 4) is a th root of unity III QUADRATIC EXPRESSION FOR THE CROSS-CORRELATION FUNCTION The cross-correlation between is given as (3) Let If is expressed in terms of a basis of over as, where, then can be easily represented as the quadratic form over,, where Note that is nothing but when in are replaced by, respectively Thus, is also a quadratic form with the coefficients From Lemma, Corollary 3, Lemma 4, in order to evaluate the exponential sum, we have to find the rank of the quadratic forms, ie, the number of solutions of the equations satisfying for all Lemma 6: The number of solutions, such that for all, equals the number of solutions satisfying (6) (7) where Let be the function defined by (4) the number of solutions, such that for all, equals the number of solutions satisfying (8) Then, the cross-correlation can be expressed as Exactly the half of the elements in are squares the other half are nonsquares Using,wehave Thus, we can represent the squares as nonsquares as, where is a nonsquare in Also, note that as runs through, each, either a square or a nonsquare appears twice Hence, we can express as where is a nonsquare in Proof: The equation Then, (10) can be rewritten as Hence, (11) holds for all (9) can be written as (10) (11) only (5) (8) are satisfied simultaneously (1)

4 1876 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 3, MARCH 01 Next, we will show that all solutions satisfying (8) also satisfy (1) From (8), we have Let Then, (17) can be rewritten as raising to the power gives Hence, we have (13) (18) Using (13), (1) can be rewritten as Since is an element in, (18) can be rewritten as (19) Since is a nonsquare in, is not in Thus, we have (19) can be rewritten as From (0), we have (0) where Hence, we only need to calculate the number of solutions for (8) to determine the number of solutions for The case of can be proved similarly From Lemma 6, we have to find out the number of solutions of (8) (9) to find the rank of, respectively Since the degrees of (8) (9) are both they are both linearized forms, the possible number of solutions for both equations is or Now, we will use the following lemma to show that at least one of the two equations,, has just one solution Lemma 7: When is a nonsquare in, the equation has as its only solution in Proof: Since it is trivial that is a solution, we have to show that has no solution in, when is a nonsquare in The previous equation can be rewritten as (14) (15) The right-h side of (15) is expressed as, which is an element in Thus, in order to prove the lemma, we should show (16) Now, we are ready to show that there are no such satisfying the previous equation Since, (1) implies that (1) In other words, (1) implies that is in But, this is not true since is not in, while is in Hence, (18) does not hold the proof is completed Clearly, either or is a nonsquare Thus, at least one of the two, ie, (8) (9), has a single solution Also, we can show that always has a single solution Lemma 8: When, ie,, () has as its only solution in, where is a nonsquare in Proof: Since we have is an odd integer, any nonsquare satisfies where is a nonsquare in Suppose that there exists, such that From, () can be rewritten as Then, we have (17) Since, for some But there is no such in satisfying, since does not divide any odd multiples of

5 CHOI et al: ON THE CROSS-CORRELATION OF A -ARY -SEQUENCE OF PERIOD 1877 The discussion on the ranks of so far can be summarized as follows Corollary 9: The possible ranks of are as follows Case 1 is a square in or has one solution, has solutions, has solutions Case is a nonsquare in has one solution, has solutions, has solutions, where denote the ranks of, respectively, IV UPPER BOUND ON CROSS-CORRELATION MAGNITUDES In this section, the upper bound on the magnitude of the crosscorrelation function of -ary -sequence its decimated sequences in (3) will be derived Theorem 10: Let, be an odd prime,, where is an odd integer Then, the magnitude of in (3) is upper bounded by In this case, when, wehave when,wehave Case 3 Rank of rank of (or rank of rank of ) From Corollary 5, we have (5) We also have Hence, the magnitude of is upper bounded by From Theorem 10, (3), (4), (5), the possible values of the cross-correlation function are given as follows 1) For, ) For ( ), Proof: Using in (7), (6) can be rewritten as 3) For, Recall that both are the quadratic forms is a nonsquare in Let be the values defined in Lemma 4 corresponding to the quadratic forms of, respectively Due to Corollary 9, the following three cases should be considered to determine the value of Case 1 Rank of rank of From Corollary 5, we have (3) Thus, we obtain Case Rank of rank of (or rank of rank of ) From Corollary 5, we have In each case, there are two pairs of the correlation values occurring the same number of times Taking this fact into account, deriving the exact distribution of cross-correlation values, ie, the number of occurrences of each correlation value requires seven independent equations for nine equations otherwise, which does not seem to be an easy task We leave this as a future work V CONSTRUCTION OF A NEW FAMILY In this section, we construct a family of -ary sequences of period with low correlation using a -ary -sequence its distinct decimated sequences defined in (1) An immediate cidate for the family is a Gold-like sequence family Let be the sequence family defined by (4)

6 1878 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 3, MARCH 01 TABLE II COMPARISON WITH GOLD-LIKE SEQUENCE FAMILIES * p is a prime number For two sequences in the family, the correlation function between the two sequences is expressed as Define the set as Then, for two sequences in the family, such that,we have (6) From (6) Theorem 10, it can be easily checked that except for, the correlation function is upper bounded by Now, let us check the case when For the case when, in (6) can be rewritten as The evaluation of requires the following lemma Lemma 11 ([]): Let be an odd prime,, suppose for some integer Suppose that is the least integer such that 1) For an even, an odd, an odd, ) For all other cases,,, Now, let us apply Lemma 11 to our case Since, set Then, which far exceeds Hence, in order to obtain the sequence family with the maximum correlation magnitude of, we have to devise a method of excluding the sequences in satisfying from Note that can be partitioned into for even odd, respectively Let, The set has an interesting property stated in the following lemmas Lemma 1: Let be an odd prime is closed under addition Proof: Let for integers Then, we have Since, Lemma 13: Let be an odd prime Proof: Since,it is enough to show that all the elements, are distinct Suppose that, where Then, Lemma 1 implies that, ie, By finding a proper set, such that we can construct a sequence family with

7 CHOI et al: ON THE CROSS-CORRELATION OF A -ARY -SEQUENCE OF PERIOD 1879 the maximum magnitude as in the following theorem of cross-correlation values Theorem 14: Let, where is an odd integer Let, Let Let be the sequence family defined by Then, the magnitude of correlation values of two sequences in is upper bounded by its family size is Proof: It is enough to show that for : where Therefore, from Lemmas 1 13, we can easily see that, thus, Some known sequence families with low correlation are listed in Table II VI CONCLUSION In this paper, we have derived the upper bound on the magnitudes on cross-correlation values of a -ary -sequence its decimation sequences for The upper bound is equal to We have also constructed the sequence family by using the same decimation value, where the family size is the magnitude of correlation values is upper bounded by ACKNOWLEDGMENT The authors would like to thank the associate editor the anonymous reviewers for their valuable comments suggestions that greatly improved the quality of this paper REFERENCES [1] H M Trachtenberg, On the cross-correlation functions of maximal recurring sequences, PhD dissertation, Univ Southern Calornia, Los Angeles, CA, 1970 [] T Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math, vol 16, pp 09 3, 1976 [3] H Dobbertin, T Helleseth, P V Kumar, H Martinsen, Ternary m-sequences with three-valued cross-correlation function: New decimations of Welch Niho type, IEEE Trans Inf Theory, vol 47, no 4, pp , May 001 [4] H Dobbertin, P Felke, T Helleseth, P Rosendahl, Niho type cross-correlation functions via Dickson polynomials Kloosterman sums, IEEE Trans Inf Theory, vol 5, no, pp , Feb 006 [5] G J Ness, T Helleseth, A Kholosha, On the correlation distribution of the Coulter Matthews decimation, IEEE Trans Inf Theory, vol 5, no 5, pp 41 47, May 006 [6] P V Kumar O Moreno, Prime-phase sequences with periodic correlation properties better than binary sequences, IEEE Trans Inf Theory, vol 37, no 3, pp , May 1991 [7] E N Müller, On the cross-correlation of sequences over GF (p) with short periods, IEEE Trans Inf Theory, vol 45, no 1, pp 89 95, Jan 1999 [8] Z Hu, X Li, D Mills, E Muller, W Sun, W Williems, Y Yang, Z Zhang, On the cross-correlation of sequences with the decimation factor d = 0, Applicable Algebra Eng Commun Comput, vol 1, pp 55 63, 001 [9] R Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans Inf Theory, vol 14, no 1, pp , Jan 1968 [10] J-S No P V Kumar, A new family of binary pseudorom sequences having optimal periodic correlation properties large linear span, IEEE Trans Inf Theory, vol 35, no, pp , Mar 1989 [11] J D Olsen, R A Scholtz, L R Welch, Bent-function sequences, IEEE Trans Inf Theory, vol 8, no 6, pp , Nov 198 [1] E Y Seo, Y S Kim, J S No, D J Shin, Cross-correlation distribution of p-ary m-sequence of period p 0 1 its decimated sequences by ( ), IEEE Trans Inf Theory, vol 54, no 7, pp , Jul 008 [13] E-Y Seo, Y-S Kim, J-S No, D-J Shin, Cross-correlation distribution of p-ary m-sequence its p +1decimated sequences with shorter period, IEICE Trans Fund Electron, Commun Comput Sci, vol E90-A, no 11, pp , Nov 007 [14] J-W Jang, Y-S Kim, J-S No, T Helleseth, New family of p-ary sequences with optimal correlation property large linear span, IEEE Trans Inf Theory, vol 50, no 8, pp , Aug 004 [15] R Lidl H Niederreiter, Finite Fields Reading, MA: Addison- Wesley, 1983, vol 0, Encyclopedia of Mathematics Its Applications [16] S T Choi, T Lim, J S No, H Chung, On the cross-correlation of a ternary m-sequence of period 3 01 its decimated sequences by, in Proc IEEE Int Symp Inf Theory, Austin, TX, Jun 010, pp [17] Y S Kim, J S Chung, J S No, D J Shin, New families of p-ary sequences with good correlation large linear complexity, IEEE Trans Inf Theory, submitted for publication Sung-Tai Choi received the BS degree in electrical engineering computer science from Seoul National University, Seoul, Korea, in 006, where he is currently working toward the PhD degree in electrical engineering computer science His area of research interests includes pseudorom sequences, wireless communication, error-correcting codes, cyptography Taehyung Lim received the BS MS degrees in electrical engineering from Seoul National University, Seoul, Korea, in , respectively, is currently pursuing the PhD degree in electrical computer engineering, University of Calornia, San Diego His research interests include pseudorom sequences, error correcting codes, communication theory Jong-Seon No (S 80 M 88 SM 10 F 1) received the BS MSEE degrees in electronics engineering from Seoul National University, Seoul, Korea, in , respectively, the PhD degree in electrical engineering from the University of Southern Calornia, Los Angeles, in 1988 He was a Senior MTS at Hughes Network Systems from February 1988 to July 1990 He was also an Associate Professor in the Department of Electronic Engineering, Konkuk University, Seoul, Korea, from September 1990 to July 1999 He joined the faculty of the Department of Electrical Engineering Computer Science, Seoul National University, in August 1999, where he is currently a Professor His research interests include error-correcting codes, sequences, cryptography, space-time codes, LDPC codes, wireless communication systems Habong Chung (S 86 M 89) received the BS degree from Seoul National University, Seoul, in 1981, the MS the PhD degrees from the University of Southern Calornia, Los Angeles, in , respectively From 1988 to 1991, he was an Assistant Professor in the Department of Electrical Computer Engineering, the State University of New York at Buffalo Since 1991, he has been with the School of Electronic Electrical Engineering, Hongik University, Seoul, where he is a Professor His research interests include coding theory, combinatorics, sequence design