Signal Sets From Functions With Optimum Nonlinearity
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 5, MAY 2007
Signal Sets From Functions With Optimum Nonlinearity Cunsheng Ding and Jianxing Yin
Abstract—Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions ) signal and almost bent functions are employed to obtain ( 2 + sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor. Index Terms—Almost bent functions, codebooks, mutually unbiased bases, maximum-Welch-bound-equality (MWBE) codebooks, planar functions, signal sets, Welch bounds.
Lemma 1: (Welch’s bounds) For any , then
signal set with
(1) , where with equality if and only if denotes the identity matrix. We have also (2) with equality if and only if for all pairs
with
to yield (3)
I. INTRODUCTION
L
ET be a set, where each is a unit complex vector over an alphabet . Such a norm 1 signal set (also called codebook). The set is called an root-mean-square (rms) correlation and the maximum correlation amplitudes of such a signal set are defined as
where is the Hermite transpose of the 1 complex vector , and is the standard inner product. We have the following bounds [19], [25]. Paper approved by R. Schober, the Editor for Detection, Equalization and MIMO of the IEEE Communications Society. Manuscript received January 2, 2006; revised May 21, 2006 and August 24, 2006. This work was supported by the Research Grants Council of the Hong Kong Special Administration Region, China, and the Natural Science Foundation of China under Project NSFC 10671140. C. Ding is with the Department of Computer Science, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China (e-mail: [email protected]). J. Yin is with the Department of Mathematics, Suzhou University, Suzhou, 215006, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2007.894113
A proof of the following lemma can be found in [23]. Lemma 2: No real signal set can meet the Welch bound of (2), if . signal set can meet the Welch bound of (2), if No . is referred to as a If the equality holds in (1), then Welch-bound-equality (WBE) signal set. A signal set meeting the bound of (2) is called a maximum-Welch-bound-equality (MWBE) signal set. An MWBE signal set must be a WBE signal set, but a WBE signal set may not be an MWBE signal set. MWBE signal sets form a subset of WBE signal sets. It is very easy to construct WBE signal sets. Every linear error-correcting code whose dual code with Hamming distance of at least three yields a WBE signal set (see [19] and [22]). The reader is referred to [22] for detailed information on the construction of WBE signal sets. MWBE signal sets are very hard to construct, as pointed out in [22, p. 100]. The known classes of MWBE signal sets are the and MWBE signal sets [4], [23], [27], trivial [22] and the following: MWBE signal sets based on conference ma• The and is a positive trices [4], [23], when integer, and with a prime number and a positive integer; and simplex signal sets. MWBE signal sets based on difference sets [27], • [8]. It is noted that Welch’s bound on the maximum correlation amplitude cannot be achieved in certain cases either (see Lemma 2), as it is not tight in these cases. The following bounds developed by Levenstein are better than Welch’s bound in certain cases [14], [17].
0090-6778/$25.00 © 2007 IEEE
DING AND YIN: SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY
For any have
real signal set
with
, we
(4) For any
complex signal set
with
, we have (5)
Another lower bound documented in [27] is (6) If (respectively ), the Welch bound on real (complex) signal sets is the best among the three bounds. However, the Levenstein bound of (4) on real signal sets and the Levenis tighter than Welch’s bound if stein bound of (5) on complex signal sets is tighter than Welch’s . The bound of (6) could be negative, e.g., when bound if and , and thus makes no sense in certain cases. However, in some cases, the lower bound of (6) could be tighter, compared with the Welch and Levenstein bounds. In code-division multiplex-access (CDMA) applications, a signal set is used to distinguish between the signals of different of users. To this end, the maximum correlation amplitude the signal set should be as small as possible. Direct-sequence CDMA (DS-CDMA) is one of the approaches to spread spectrum for signal transmission. Optimal and almost optimal signal sets with are desirable in synchronous DS-CDMA systems for reducing interference, where the number of users is greater than the signal space dimension or the spreading factor . Such signal sets could be quite useful in third-generation (3G) and fourth-generation (4G) networks for increasing the subscriber capacity within the limited spectral resource. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obsignal sets. The signal sets derived from planar tain functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets presented in this paper could have a very small alphabet size and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor. II. PLANAR FUNCTIONS Let be a function from a finite abelian group to an. We say that is linear if and other finite abelian group for all . A function only if is affine if and only if , where is linear and is a constant. A robust measure of nonlinearity of is defined by
where value of
denotes the cardinality of the set . The smaller the , the higher the corresponding nonlinearity of .
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[3]. A function It is easily seen that has perfect nonlinearity if . A perfect nonlinear function from a finite abelian group to a finite abelian group of the same order is called a planar function in finite geometry. Planar functions were introduced by Dembowski and Ostrom in 1968 for the construction of affine planes [6]. Thus, perfect nonlinear functions introduced by Nyberg [20] in 1993 and bent functions introduced by Rothaus [21] in 1976 are just extensions of planar functions. We refer to Carlet and Ding [3] for a survey of highly nonlinear functions and Coulter and Matthews [5] and Ding and Yuan [9] for information about planar functions. . Known perfect nonLet be an odd prime, and let to are equivalent to one of linear functions from the following [3], [5]: ; • • , where is odd [6]; • , where , is odd, and [5]; , where and is odd • and [9] for the general case). (see [5] for the case Note that planar functions over exist for any pair where is an odd prime number. III. CONSTRUCTION OF SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY Throughout this section, a positive integer. We use elements of the finite field . Let define on . Define
, where is a prime and is to denote all of the . For any positive integer , denote the absolute trace function (7)
Then is an additive group character of In this section, we define
.
...... ...
which are mutually orthogonal, have the unit-norm, and form the standard basis of the -dimensional Hilbert space. We use to denote this set of unit-norm vectors. Let be a function from to . For each , we define the unit-norm vector pair
Then, we define the signal set (8) The signal set is optimal or nearly optimal with respect to the Levenstein bounds if has optimum nonlinearity. In the sequel, we consider this construction by separating the case from the case .
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A. Optimum Complex Signal Sets From Planar Functions Here, we assume that is an odd prime, so that there are to . We first introduce planar functions the following lemma, which is a special case of [3, Theorem 16]. For completeness, we give a simpler proof here. to Lemma 3: Let be a planar function from . Then, we have
. Then, 1) Example 1: Take the planar function over the signal set given by consists of the following nine signals:
where , , and denote , respectively.
,
, and
B. Nearly Optimum Signal Sets From Almost Bent Functions Note that there is no planar function from . For any function from to define
Proof: Define
to , we
Then, it follows from the definition of planar functions that A function
This completes the proof. Theorem 4: Let be a planar function from to . Then, is a signal set with . Proof: Note that is planar for any and . It follows from Lemma 3 that, for any distinct and , we have
if if
from to is called almost bent if , or for every pair with . Let be odd. Known almost bent functions are the following. (Gold function), [10]. 1) (Kasami function), 2) [16]. 3) (Welch function) [16]. (Niho function), 4) [12], [13]. (Niho function), 5) [12], [13]. Hence, almost bent functions over exist for every odd . be an almost bent function from Theorem 5: Let to . Then, is a signal set with . Proof: Note that for any , and, for any distinct and , we have
.
whenever This also proves that . Clearly, for any and , we have . The conclusion on then follows. This completes the proof. Take and . The Levenstein bound of (5) then becomes
Thus, the signal set of Theorems 4 meets the Levenstein bound and is thus optimal. Now we present one example of the signal sets of Theorem 4.
This also proves that . Clearly, for any
whenever and , we have
. The conclusion on then follows. This completes the proof. The construction of the signal set of Theorem 5 produced by the almost bent (Gold) function can be explained in more engineering terms as follows. We take all cyclic shifts of all Gold sequences, coming to set of signals. We then append the left symbol to all of them and add to the set one signal consisting of plus ones and all signals
DING AND YIN: SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY
of the standard basis. Then, the set obtained is the signal set of Theorem 5 when is the Gold function. Take . The Levenstein bound of (4) then becomes
Thus, the signal set of Theorem 5 nearly meets the Levenstein bound. signal sets with There are and alphabet size 5 (see [1] and [26]). They are slightly better than those of Theorem 5 in terms of the maximum correlation amplitude. However, the signal sets of Theorem 5 have alphabet size 4, and the coordinates of the signals are real-valued. A small alphabet size is of importance in applications, according to Sarwate [22]. To the best of our knowledge, the signal sets of Theorem 5 are the best known with alphabet size 4. Now, we present one example of the signal sets of Theorem 5. over . 1) Example 2: Take the Kasami function Then, the signal set of Theorem 5 given by consists of the following 72 signals:
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TABLE I EXAMPLES OF PARAMETERS OF SIGNAL SETS PRODUCED IN THIS PAPER
IV. SCALABILITY ISSUE REGARDING THE SIGNAL SETS One question regarding the signal sets of Theorem 4 (respectively, Theorem 5) is whether they are still optimal (nearly opuser signal set is constructed timal) after a users left the system, where is a posand itive integer. To answer this question, we consider the signal set of The. If users left the system, orem 4 and assume that signal set. Since then the system is reduced to a , the Levenstein bound is tighter than the Welch bound. For any signal set, the Levenstein bound becomes
On the other hand, will remain the same as before as the users who left the system could be any group of users in the users. Hence, the system will not be optimal original set of anymore if users left the system and remains the same as before. For the signal sets of Theorem 5, one can similarly prove that and the Levenstein bound increases the distance between after a user signal set is constructed and users left the system. V. CONCLUSION The construction of this paper produces infinitely many signal sets with different alphabet sizes. Table I gives examples of the parameters of the signal sets produced in this paper. Two orthogonal bases and of the -dimensional Hilbert space are called mutually unbiased if the norm of the inner equals for all and . Mutually product unbiased bases have important applications in quantum physics [26]. Define for each
where
and
denote
and
, respectively.
where is a planar function from to . Then, those together with the standard basis form a set of mutually unbiased bases. If , then it becomes the set of mutually unbiased bases obtained by Wootters and Fields [26]. Thus, the generic construction of this paper is a generalization of the construction of Wootters and Fields [26]. Note that we obtained many sets of other mutually unbiased bases by
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 5, MAY 2007
using the planar functions , , and described in Section II. The contributions of this paper include many mutually unbiased bases and the generic construction of signal sets together with specific signal sets that are based on planar and almost bent functions. Within the framework of the generic construction of yields new opthis paper, any new planar function on timum signal set meeting the Levenstein bound. is very Constructing optimal signal sets with minimal difficult in general. This problem is equivalent to line packing in Grassmannian space [4], [23]. In frame theory, such a signal minimized is referred to as a Grassmannian frame set with [23]. The signal sets presented in this paper and the constructions should have applications in these areas. Waldron [24] generalized the Welch bound and showed that those meeting the generalized Welch bound are tight frames. The performance comparison of MWBE signal sets and signal sets from mutually unbiased bases is given in [11]. It is known that Welch’s bounds are not tight for binary signal sets in several cases. Karystinos and Pados [15] have developed new bounds that improve Welch’s bound of (1) in these cases. Binary signal sets meeting Welch’s and Karystinos-Pados’ bounds are treated in [15] and [7]. ACKNOWLEDGMENT The authors wish to thank the three reviewers for their constructive comments and suggestions that improved both the quality and the presentation of this paper. Special thanks go to one of the reviewers for pointing out [14] and [17] and the Levenstein bounds. REFERENCES [1] A. R. Calderbank, P. J. Cameron, W. M. Kantor, and J. J. Seidel, “Z -kerdock codes, orthogonal spreads, and extremal Euclidean line-sets,” Proc. London Math. Soc., vol. 75, no. 3, pp. 436–480, 1997. [2] C. Carlet and S. Dubuc, “On generalized bent and q -ary perfect nonlinear functions,” in Finite Fields and Applications, Proceedings of Fq5, D. Jungnickel and H. Niederreiter, Eds. Berlin, Germany: Springer-Verlag, 2000, pp. 81–94. [3] C. Carlet and C. Ding, “Highly nonlinear mappings,” J. Complexity, vol. 20, no. 2, pp. 205–244, 2004. [4] J. H. Conway, R. H. Harding, and N. J. A. Sloane, “Packing lines, planes, etc.: Packings in Grassmannian spaces,” Exper. Math., vol. 5, no. 2, pp. 139–159, 1996. [5] R. S. Coulter and R. W. Matthews, “Planar functions and planes of Lenz–Barlotti class II,” Designs, Codes Cryptogr., vol. 10, pp. 167–184, 1997. [6] P. Dembowski and T. G. Ostrom, “Planes of order n with collineation groups of order n ,” Math. Z., vol. 193, pp. 239–258, 1968. [7] C. Ding, M. Golin, and T. Kløve, “Meeting the Welch and KarystinosPados bounds on DS-CDMA binary signature sets,” Designs, Codes Cryptogr., vol. 30, pp. 73–84, 2003. [8] C. Ding, “Complex codebooks from combinatorial designs,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4229–4235, Sep. 2006. [9] C. Ding and J. Yuan, “A family of skew Paley–Hadamard difference sets,” J. Comb. Theory A, vol. 113, no. 7, pp. 1219–1592, Oct. 2006. [10] R. Gold, “Maximal recursive sequences with 3-valued recursive crosscorrelation functions,” IEEE Trans. Inf. Theory, vol. IT-14, no. 1, pp. 154–156, Jan. 1968. [11] R. W. Heath Jr., T. Strohmer, and A. J. Paulraj, “On quasi-orthogonal signature for CDMA systems,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1217–1226, Mar. 2006. [12] H. Hollmann and Q. Xiang, “A proof of the Welch and Niho conjectures on crosscorrelations of binary m-sequences,” Finite Fields Applicat., vol. 7, pp. 253–286, 2001.
[13] X. D. Hou, “A note on the proof of Niho’s conjecture,” SIAM J. Discrete Math., vol. 18, no. 2, pp. 313–319, 2004. [14] G. A. Kabatyanskii and V. I. Levenstein, “Bounds for packing on a sphere and in space,” Probl. Inf. Transmission, vol. 14, pp. 1–17, 1978. [15] G. N. Karystinos and D. A. Pados, “New bounds on the total-squaredcorrelation and perfect design of DS-CDMA binary signature sets,” IEEE Trans. Commun., vol. 51, no. 1, pp. 48–51, Jan. 2003. [16] T. Kasami, “The weight enumerators for several classes of subcodes of the second order binary Reed–Muller codes,” Inf. Control, vol. 18, pp. 369–394, 1971. [17] V. I. Levenstein, “Bounds for packings of metric spaces and some of their applications,” (in Russian) Probl. Cybern., vol. 40, pp. 43–110, 1983. [18] D. J. Love, R. W. Heath, and T. Strohmer, “Grassmannian meamingforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003. [19] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code-division multiple-access systems,” in Sequences II: Methods in Communication, Security and Computer Science. Heidelberg, Germany: Springer, 1993, pp. 63–78. [20] K. Nyberg, “Perfect nonlinear S-boxes,” in Proc. Eurocrypt’91, Berlin, Germany, 1992, vol. 547, Lecture Notes in Computer Science, pp. 378–386. [21] O. S. Rothaus, “On bent functions,” J. Comb. Theory A, vol. 20, pp. 300–305, 1976. [22] D. Sarwate, “Meeting the Welch bound with equality,” in Proc. SETA, 1999, pp. 79–102. [23] T. Strohmer and R. W. Heath Jr., “Grassmannian frames with applications to coding and communication,” Appl. Computat. Harmonic Anal., vol. 14, no. 3, pp. 257–275, 2003. [24] S. Waldron, “Generalized Welch bound equality sequences are tight frames,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2307–2309, Sep. 2003. [25] L. Welch, “Lower bounds on the maximum cross correlation of signals,” IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 397–399, May 1974. [26] W. Wootters and B. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys., vol. 191, no. 2, pp. 363–381, 1989. [27] P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the Welch bound with difference sets,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1900–1907, May 2005.
Cunsheng Ding was born in 1962 in Shaanxi, China. He received the M.Sc. degree from the Northwestern Telecommunications Engineering Institute, Xian, China, in 1988, and the Ph.D. degree from the University of Turku, Turku, Finland, in 1997. From 1988 to 1992, he was a Lecturer of Mathematics with Xidian University, China. Before joining the Hong Kong University of Science and Technology, Kowloon, in 2000, where he is currently an Associate Professor of computer science, he was an Assistant Professor of computer science with the National University of Singapore. His research fields are cryptography and coding theory. He has coauthored four research monographs. He was the Guest Editor/Coeditor for three special issues on cryptography for Information and Computation, Theoretical Computer Science, and Journal of Complexity; and an Editor for the Journal of Communications and Networks and the Journal of Universal Computer Science. Dr. Ding was the corecipient of the State Natural Science Award of China in 1992.
Jianxing Yin graduated from Suzhou University, Suzhou, China, in 1977. In 1977, he became a Teacher with the Department of Mathematics, Suzhou University, where he is currently a Full Professor. His research interests include applications of combinatorial designs in coding theory and cryptography.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 5, MAY 2007
Signal Sets From Functions With Optimum Nonlinearity Cunsheng Ding and Jianxing Yin
Abstract—Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions ) signal and almost bent functions are employed to obtain ( 2 + sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor. Index Terms—Almost bent functions, codebooks, mutually unbiased bases, maximum-Welch-bound-equality (MWBE) codebooks, planar functions, signal sets, Welch bounds.
Lemma 1: (Welch’s bounds) For any , then
signal set with
(1) , where with equality if and only if denotes the identity matrix. We have also (2) with equality if and only if for all pairs
with
to yield (3)
I. INTRODUCTION
L
ET be a set, where each is a unit complex vector over an alphabet . Such a norm 1 signal set (also called codebook). The set is called an root-mean-square (rms) correlation and the maximum correlation amplitudes of such a signal set are defined as
where is the Hermite transpose of the 1 complex vector , and is the standard inner product. We have the following bounds [19], [25]. Paper approved by R. Schober, the Editor for Detection, Equalization and MIMO of the IEEE Communications Society. Manuscript received January 2, 2006; revised May 21, 2006 and August 24, 2006. This work was supported by the Research Grants Council of the Hong Kong Special Administration Region, China, and the Natural Science Foundation of China under Project NSFC 10671140. C. Ding is with the Department of Computer Science, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China (e-mail: [email protected]). J. Yin is with the Department of Mathematics, Suzhou University, Suzhou, 215006, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2007.894113
A proof of the following lemma can be found in [23]. Lemma 2: No real signal set can meet the Welch bound of (2), if . signal set can meet the Welch bound of (2), if No . is referred to as a If the equality holds in (1), then Welch-bound-equality (WBE) signal set. A signal set meeting the bound of (2) is called a maximum-Welch-bound-equality (MWBE) signal set. An MWBE signal set must be a WBE signal set, but a WBE signal set may not be an MWBE signal set. MWBE signal sets form a subset of WBE signal sets. It is very easy to construct WBE signal sets. Every linear error-correcting code whose dual code with Hamming distance of at least three yields a WBE signal set (see [19] and [22]). The reader is referred to [22] for detailed information on the construction of WBE signal sets. MWBE signal sets are very hard to construct, as pointed out in [22, p. 100]. The known classes of MWBE signal sets are the and MWBE signal sets [4], [23], [27], trivial [22] and the following: MWBE signal sets based on conference ma• The and is a positive trices [4], [23], when integer, and with a prime number and a positive integer; and simplex signal sets. MWBE signal sets based on difference sets [27], • [8]. It is noted that Welch’s bound on the maximum correlation amplitude cannot be achieved in certain cases either (see Lemma 2), as it is not tight in these cases. The following bounds developed by Levenstein are better than Welch’s bound in certain cases [14], [17].
0090-6778/$25.00 © 2007 IEEE
DING AND YIN: SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY
For any have
real signal set
with
, we
(4) For any
complex signal set
with
, we have (5)
Another lower bound documented in [27] is (6) If (respectively ), the Welch bound on real (complex) signal sets is the best among the three bounds. However, the Levenstein bound of (4) on real signal sets and the Levenis tighter than Welch’s bound if stein bound of (5) on complex signal sets is tighter than Welch’s . The bound of (6) could be negative, e.g., when bound if and , and thus makes no sense in certain cases. However, in some cases, the lower bound of (6) could be tighter, compared with the Welch and Levenstein bounds. In code-division multiplex-access (CDMA) applications, a signal set is used to distinguish between the signals of different of users. To this end, the maximum correlation amplitude the signal set should be as small as possible. Direct-sequence CDMA (DS-CDMA) is one of the approaches to spread spectrum for signal transmission. Optimal and almost optimal signal sets with are desirable in synchronous DS-CDMA systems for reducing interference, where the number of users is greater than the signal space dimension or the spreading factor . Such signal sets could be quite useful in third-generation (3G) and fourth-generation (4G) networks for increasing the subscriber capacity within the limited spectral resource. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obsignal sets. The signal sets derived from planar tain functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets presented in this paper could have a very small alphabet size and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor. II. PLANAR FUNCTIONS Let be a function from a finite abelian group to an. We say that is linear if and other finite abelian group for all . A function only if is affine if and only if , where is linear and is a constant. A robust measure of nonlinearity of is defined by
where value of
denotes the cardinality of the set . The smaller the , the higher the corresponding nonlinearity of .
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[3]. A function It is easily seen that has perfect nonlinearity if . A perfect nonlinear function from a finite abelian group to a finite abelian group of the same order is called a planar function in finite geometry. Planar functions were introduced by Dembowski and Ostrom in 1968 for the construction of affine planes [6]. Thus, perfect nonlinear functions introduced by Nyberg [20] in 1993 and bent functions introduced by Rothaus [21] in 1976 are just extensions of planar functions. We refer to Carlet and Ding [3] for a survey of highly nonlinear functions and Coulter and Matthews [5] and Ding and Yuan [9] for information about planar functions. . Known perfect nonLet be an odd prime, and let to are equivalent to one of linear functions from the following [3], [5]: ; • • , where is odd [6]; • , where , is odd, and [5]; , where and is odd • and [9] for the general case). (see [5] for the case Note that planar functions over exist for any pair where is an odd prime number. III. CONSTRUCTION OF SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY Throughout this section, a positive integer. We use elements of the finite field . Let define on . Define
, where is a prime and is to denote all of the . For any positive integer , denote the absolute trace function (7)
Then is an additive group character of In this section, we define
.
...... ...
which are mutually orthogonal, have the unit-norm, and form the standard basis of the -dimensional Hilbert space. We use to denote this set of unit-norm vectors. Let be a function from to . For each , we define the unit-norm vector pair
Then, we define the signal set (8) The signal set is optimal or nearly optimal with respect to the Levenstein bounds if has optimum nonlinearity. In the sequel, we consider this construction by separating the case from the case .
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 5, MAY 2007
A. Optimum Complex Signal Sets From Planar Functions Here, we assume that is an odd prime, so that there are to . We first introduce planar functions the following lemma, which is a special case of [3, Theorem 16]. For completeness, we give a simpler proof here. to Lemma 3: Let be a planar function from . Then, we have
. Then, 1) Example 1: Take the planar function over the signal set given by consists of the following nine signals:
where , , and denote , respectively.
,
, and
B. Nearly Optimum Signal Sets From Almost Bent Functions Note that there is no planar function from . For any function from to define
Proof: Define
to , we
Then, it follows from the definition of planar functions that A function
This completes the proof. Theorem 4: Let be a planar function from to . Then, is a signal set with . Proof: Note that is planar for any and . It follows from Lemma 3 that, for any distinct and , we have
if if
from to is called almost bent if , or for every pair with . Let be odd. Known almost bent functions are the following. (Gold function), [10]. 1) (Kasami function), 2) [16]. 3) (Welch function) [16]. (Niho function), 4) [12], [13]. (Niho function), 5) [12], [13]. Hence, almost bent functions over exist for every odd . be an almost bent function from Theorem 5: Let to . Then, is a signal set with . Proof: Note that for any , and, for any distinct and , we have
.
whenever This also proves that . Clearly, for any and , we have . The conclusion on then follows. This completes the proof. Take and . The Levenstein bound of (5) then becomes
Thus, the signal set of Theorems 4 meets the Levenstein bound and is thus optimal. Now we present one example of the signal sets of Theorem 4.
This also proves that . Clearly, for any
whenever and , we have
. The conclusion on then follows. This completes the proof. The construction of the signal set of Theorem 5 produced by the almost bent (Gold) function can be explained in more engineering terms as follows. We take all cyclic shifts of all Gold sequences, coming to set of signals. We then append the left symbol to all of them and add to the set one signal consisting of plus ones and all signals
DING AND YIN: SIGNAL SETS FROM FUNCTIONS WITH OPTIMUM NONLINEARITY
of the standard basis. Then, the set obtained is the signal set of Theorem 5 when is the Gold function. Take . The Levenstein bound of (4) then becomes
Thus, the signal set of Theorem 5 nearly meets the Levenstein bound. signal sets with There are and alphabet size 5 (see [1] and [26]). They are slightly better than those of Theorem 5 in terms of the maximum correlation amplitude. However, the signal sets of Theorem 5 have alphabet size 4, and the coordinates of the signals are real-valued. A small alphabet size is of importance in applications, according to Sarwate [22]. To the best of our knowledge, the signal sets of Theorem 5 are the best known with alphabet size 4. Now, we present one example of the signal sets of Theorem 5. over . 1) Example 2: Take the Kasami function Then, the signal set of Theorem 5 given by consists of the following 72 signals:
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TABLE I EXAMPLES OF PARAMETERS OF SIGNAL SETS PRODUCED IN THIS PAPER
IV. SCALABILITY ISSUE REGARDING THE SIGNAL SETS One question regarding the signal sets of Theorem 4 (respectively, Theorem 5) is whether they are still optimal (nearly opuser signal set is constructed timal) after a users left the system, where is a posand itive integer. To answer this question, we consider the signal set of The. If users left the system, orem 4 and assume that signal set. Since then the system is reduced to a , the Levenstein bound is tighter than the Welch bound. For any signal set, the Levenstein bound becomes
On the other hand, will remain the same as before as the users who left the system could be any group of users in the users. Hence, the system will not be optimal original set of anymore if users left the system and remains the same as before. For the signal sets of Theorem 5, one can similarly prove that and the Levenstein bound increases the distance between after a user signal set is constructed and users left the system. V. CONCLUSION The construction of this paper produces infinitely many signal sets with different alphabet sizes. Table I gives examples of the parameters of the signal sets produced in this paper. Two orthogonal bases and of the -dimensional Hilbert space are called mutually unbiased if the norm of the inner equals for all and . Mutually product unbiased bases have important applications in quantum physics [26]. Define for each
where
and
denote
and
, respectively.
where is a planar function from to . Then, those together with the standard basis form a set of mutually unbiased bases. If , then it becomes the set of mutually unbiased bases obtained by Wootters and Fields [26]. Thus, the generic construction of this paper is a generalization of the construction of Wootters and Fields [26]. Note that we obtained many sets of other mutually unbiased bases by
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using the planar functions , , and described in Section II. The contributions of this paper include many mutually unbiased bases and the generic construction of signal sets together with specific signal sets that are based on planar and almost bent functions. Within the framework of the generic construction of yields new opthis paper, any new planar function on timum signal set meeting the Levenstein bound. is very Constructing optimal signal sets with minimal difficult in general. This problem is equivalent to line packing in Grassmannian space [4], [23]. In frame theory, such a signal minimized is referred to as a Grassmannian frame set with [23]. The signal sets presented in this paper and the constructions should have applications in these areas. Waldron [24] generalized the Welch bound and showed that those meeting the generalized Welch bound are tight frames. The performance comparison of MWBE signal sets and signal sets from mutually unbiased bases is given in [11]. It is known that Welch’s bounds are not tight for binary signal sets in several cases. Karystinos and Pados [15] have developed new bounds that improve Welch’s bound of (1) in these cases. Binary signal sets meeting Welch’s and Karystinos-Pados’ bounds are treated in [15] and [7]. ACKNOWLEDGMENT The authors wish to thank the three reviewers for their constructive comments and suggestions that improved both the quality and the presentation of this paper. Special thanks go to one of the reviewers for pointing out [14] and [17] and the Levenstein bounds. REFERENCES [1] A. R. Calderbank, P. J. Cameron, W. M. Kantor, and J. J. Seidel, “Z -kerdock codes, orthogonal spreads, and extremal Euclidean line-sets,” Proc. London Math. Soc., vol. 75, no. 3, pp. 436–480, 1997. [2] C. Carlet and S. Dubuc, “On generalized bent and q -ary perfect nonlinear functions,” in Finite Fields and Applications, Proceedings of Fq5, D. Jungnickel and H. Niederreiter, Eds. Berlin, Germany: Springer-Verlag, 2000, pp. 81–94. [3] C. Carlet and C. Ding, “Highly nonlinear mappings,” J. Complexity, vol. 20, no. 2, pp. 205–244, 2004. [4] J. H. Conway, R. H. Harding, and N. J. A. Sloane, “Packing lines, planes, etc.: Packings in Grassmannian spaces,” Exper. Math., vol. 5, no. 2, pp. 139–159, 1996. [5] R. S. Coulter and R. W. Matthews, “Planar functions and planes of Lenz–Barlotti class II,” Designs, Codes Cryptogr., vol. 10, pp. 167–184, 1997. [6] P. Dembowski and T. G. Ostrom, “Planes of order n with collineation groups of order n ,” Math. Z., vol. 193, pp. 239–258, 1968. [7] C. Ding, M. Golin, and T. Kløve, “Meeting the Welch and KarystinosPados bounds on DS-CDMA binary signature sets,” Designs, Codes Cryptogr., vol. 30, pp. 73–84, 2003. [8] C. Ding, “Complex codebooks from combinatorial designs,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4229–4235, Sep. 2006. [9] C. Ding and J. Yuan, “A family of skew Paley–Hadamard difference sets,” J. Comb. Theory A, vol. 113, no. 7, pp. 1219–1592, Oct. 2006. [10] R. Gold, “Maximal recursive sequences with 3-valued recursive crosscorrelation functions,” IEEE Trans. Inf. Theory, vol. IT-14, no. 1, pp. 154–156, Jan. 1968. [11] R. W. Heath Jr., T. Strohmer, and A. J. Paulraj, “On quasi-orthogonal signature for CDMA systems,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1217–1226, Mar. 2006. [12] H. Hollmann and Q. Xiang, “A proof of the Welch and Niho conjectures on crosscorrelations of binary m-sequences,” Finite Fields Applicat., vol. 7, pp. 253–286, 2001.
[13] X. D. Hou, “A note on the proof of Niho’s conjecture,” SIAM J. Discrete Math., vol. 18, no. 2, pp. 313–319, 2004. [14] G. A. Kabatyanskii and V. I. Levenstein, “Bounds for packing on a sphere and in space,” Probl. Inf. Transmission, vol. 14, pp. 1–17, 1978. [15] G. N. Karystinos and D. A. Pados, “New bounds on the total-squaredcorrelation and perfect design of DS-CDMA binary signature sets,” IEEE Trans. Commun., vol. 51, no. 1, pp. 48–51, Jan. 2003. [16] T. Kasami, “The weight enumerators for several classes of subcodes of the second order binary Reed–Muller codes,” Inf. Control, vol. 18, pp. 369–394, 1971. [17] V. I. Levenstein, “Bounds for packings of metric spaces and some of their applications,” (in Russian) Probl. Cybern., vol. 40, pp. 43–110, 1983. [18] D. J. Love, R. W. Heath, and T. Strohmer, “Grassmannian meamingforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003. [19] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code-division multiple-access systems,” in Sequences II: Methods in Communication, Security and Computer Science. Heidelberg, Germany: Springer, 1993, pp. 63–78. [20] K. Nyberg, “Perfect nonlinear S-boxes,” in Proc. Eurocrypt’91, Berlin, Germany, 1992, vol. 547, Lecture Notes in Computer Science, pp. 378–386. [21] O. S. Rothaus, “On bent functions,” J. Comb. Theory A, vol. 20, pp. 300–305, 1976. [22] D. Sarwate, “Meeting the Welch bound with equality,” in Proc. SETA, 1999, pp. 79–102. [23] T. Strohmer and R. W. Heath Jr., “Grassmannian frames with applications to coding and communication,” Appl. Computat. Harmonic Anal., vol. 14, no. 3, pp. 257–275, 2003. [24] S. Waldron, “Generalized Welch bound equality sequences are tight frames,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2307–2309, Sep. 2003. [25] L. Welch, “Lower bounds on the maximum cross correlation of signals,” IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 397–399, May 1974. [26] W. Wootters and B. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys., vol. 191, no. 2, pp. 363–381, 1989. [27] P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the Welch bound with difference sets,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1900–1907, May 2005.
Cunsheng Ding was born in 1962 in Shaanxi, China. He received the M.Sc. degree from the Northwestern Telecommunications Engineering Institute, Xian, China, in 1988, and the Ph.D. degree from the University of Turku, Turku, Finland, in 1997. From 1988 to 1992, he was a Lecturer of Mathematics with Xidian University, China. Before joining the Hong Kong University of Science and Technology, Kowloon, in 2000, where he is currently an Associate Professor of computer science, he was an Assistant Professor of computer science with the National University of Singapore. His research fields are cryptography and coding theory. He has coauthored four research monographs. He was the Guest Editor/Coeditor for three special issues on cryptography for Information and Computation, Theoretical Computer Science, and Journal of Complexity; and an Editor for the Journal of Communications and Networks and the Journal of Universal Computer Science. Dr. Ding was the corecipient of the State Natural Science Award of China in 1992.
Jianxing Yin graduated from Suzhou University, Suzhou, China, in 1977. In 1977, he became a Teacher with the Department of Mathematics, Suzhou University, where he is currently a Full Professor. His research interests include applications of combinatorial designs in coding theory and cryptography.
