A Class of Optimal Frequency Hopping Sequences with ... - IEEE Xplore

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012

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A Class of Optimal Frequency Hopping Sequences with New Parameters Xiangyong Zeng, Han Cai, Xiaohu Tang, Member, IEEE, and Yang Yang

Abstract—In this paper, we propose an interleaving construction of new sets of frequency hopping sequences from the known ones. By choosing suitable known optimal frequency hopping sequences and sets of frequency hopping sequences and then recursively applying the proposed construction, optimal frequency hopping sequences and sets of frequency hopping sequences with new parameters can be obtained. Index Terms—Frequency hopping sequence (FHS), Hamming correlation, interleaving technique, Lempel–Greenberger bound, Peng–Fan bound.

I. INTRODUCTION

F

REQUENCY hopping multiple-access is widely used in modern communication systems such as ultrawideband, military communications, Bluetooth, and so on [21]. In those systems, we have to minimize the maximum of Hamming out-of-phase autocorrelation and cross correlation of the set of frequency hopping sequences (FHSs) to reduce the multiple-access interference. To accommodate many users, it is also very desirable that size of the FHS sets is as large as possible. However, the parameters of the FHS sets are subjected to some theoretic bounds, for example, the Lempel–Greenberger bound [18], the Peng–Fan bound [20], or the coding theory bounds [7]. Therefore, it is of great interest to construct optimal FHSs with respect to the bounds. During the decades, numerous algebraic and combinatorial constructions of optimal FHSs and FHS sets have been proposed (see [1]–[10], [13]–[18], [22]–[25], and references therein). The interleaving technique is a method to construct a long from sequences of length . They sequence of length have been widely used in constructing sequences with good periodic correlation [11], [12]. In 2010, Chung et al. introduced interleaving technique to the design of FHSs with good Hamming correlation [2]. Based on known optimal FHS sets, they presented new classes of optimal FHSs with respect to the Lempel–Greenberger bound and the Peng–Fan bound via

Manuscript received January 03, 2012; revised April 08, 2012; accepted April 10, 2012. Date of publication May 03, 2012; date of current version June 12, 2012. The work of X. Zeng and H. Cai was supported by the National Science Foundation of China (NSFC) under Grant 61170257. The work of X. Tang and Y. Yang was supported by the NSFC under Grant 61171095. X. Zeng and H. Cai are with the Faculty of Mathematics and Computer Science, Hubei University, Wuhan 430062, China (e-mail: xiangyongzeng@yahoo. com.cn; [email protected]). X. Tang and Y. Yang are with the Provincial Key Lab of Information Coding and Transmission, Institute of Mobile Communications, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]; [email protected]). Communicated by T. Helleseth, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2012.2195771

interleaving the known FHS sets. Each FHS in the new optimal FHS set constructed by their method can be arranged into a matrix such that each column of the new FHS is exactly an FHS in the corresponding known FHS set. Compared to the original one, the new set has longer sequence length, larger Hamming correlation, the same alphabet, and less number of sequences. The purpose of this paper is to present a new construction of optimal FHSs and FHS sets with new parameters by means of interleaving technique. We present a construction of FHS sets based on known ones, which results in new FHS sets having a longer sequence length, the same maximum nontrivial Hamming correlation, and the same number of sequences but a larger size of alphabet. Roughly speaking, in contrast to Chung et al.’s method, our interleaving approach improves the size of sequence sets but with larger alphabet, if applied to the same known FHS sets. In our study, some known optimal FHSs (respectively, optimal sets of FHSs) are used in the proposed construction to construct new optimal FHSs (respectively, optimal sets of FHSs). We list the parameters of some optimal FHSs and sets of FHSs obtained by the proposed construction in Section IV. Furthermore, if the parameters of these sequences satisfy some extra conditions, then they can be used to recursively construct more optimal FHSs and sets of FHSs, whose parameters have not been reported in the literature. The remainder of this paper is organized as follows. In Section II, we recall some preliminaries. A construction of FHS sets is proposed in Section III, and the properties of these FHS sets are also analyzed. In Section IV, some optimal FHSs and sets of FHSs are obtained based on known optimal FHSs and sets of FHSs, respectively. Section V concludes the study. II. PRELIMINARIES For a positive integer , let be a set of available frequencies, also called the alphabet. A sequence is called an FHS of length over if for all . For two FHSs and of length over , their Hamming correlation is defined by (1) if , and 0 otherwise, and the addiwhere tion in the subscript is performed modulo . If for all , then we say and call the Hamming autocorrelation of , denoted by for short. Define as

0018-9448/$31.00 © 2012 IEEE

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for an FHS


, and

as

for two FHSs and such that . Throughout this paper, let denote an FHS of length over the alphabet of size , with . In this case, we say that the sequence has parameters . For , let denote the times of occurring in . If for any , then is called balanced; otherwise, is called unbalanced. For a real number , let denote the smallest integer not less than and denote the integer part of . A lower bound of was established by Lempel and Greenberger as below. Lemma 1 (see [18]): For every FHS alphabet of size

of length

over an

where is the least nonnegative residue of modulo . We denote by the right-hand side of the inequality in Lemma 1. An FHS is called optimal if , that is to say, is optimal with respect to the Lempel–Greenberger bound, and it is called near-optimal if , that is to say, is near-optimal with respect to the Lempel–Greenberger bound. Usually, it is more convenient to check the optimality of an FHS with respect to the Lempel–Greenberger bound by the following proposition. Proposition 1 (see [10]): For an

FHS (2)

where . This implies that when if , then the sequence is optimal. Let be a set of FHSs of length over the alphabet size , and the maximum nontrivial Hamming correlation of the sequence set is defined by

, of

We use to denote the set with . In this case, we say that the set has parameters . The Lempel–Greenberger bound in Lemma 1 is independent of the size of an FHS set. In 2004, Peng and Fan developed the following bounds on by taking account of the number of sequences in the set . Lemma 2 (see [20]): Let be a set of over an alphabet of size . Define

FHSs of length . Then (3)

and (4)

FHS set is called optimal if one In this paper, an of the Peng–Fan bounds in Lemma 2 is met. Some known optimal FHSs and sets of FHSs are listed in Tables I and II respectively, where , , and are three positive integers, and are two primes, and is a prime power.

III. A CONSTRUCTION OF FHS SETS For a prime power , let denote the finite field with elements and be its extension field with degree , where is a positive integer [19]. Let be a primitive element of the finite field , i.e., . It is well known that there exist elements in such that the cosets

of

for

satisfy

and

when . Set , . Our construction begins with a known FHS set. To the best of our knowledge, the length of an optimal FHS or an FHS in an optimal FHS set is a prime (see [1], [3], [7], [9], [13], and [24]), a square of a prime [17], a product of two primes (see [2], [3], and [10]), a factor of for a prime and a positive integer (see [5]–[8], [10], [13], [15]–[18], [22], [24], and [25]), or an integer with a special form as those in [2], [3], and [10]. Step 1: Choose an FHS set over the alphabet with parameters satisfying 1) ; 2) for each with , where denotes the times of occurring in . Step 2: Choose different integers from the set . Step 3: For each with , define an FHS associated with the FHS of in the following way: for every with , if for then take such that (5) for any two different pairs of integers . Concerning , we have the following fact. Fact 1: 1) if for ; 2) if and only if ; if and only if and . 3) Based on the set obtained in Step 3, we present an interleaving construction of an FHS set. Construction A: An FHS set with for is defined as (6)

ZENG et al.: CLASS OF OPTIMAL FREQUENCY HOPPING SEQUENCES

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TABLE I SOME KNOWN OPTIMAL FHSS WITH PARAMETERS

Proof: Obviously for any integer with ranges over with varying from 0 to we then have

TABLE II SOME KNOWN OPTIMAL SETS OF FHSS WITH PARAMETERS

, . By (6),

The conclusion immediately follows from Fact 1.1. Lemma 4: The alphabet Proof: Given , straightforwardly

satisfies with

. and

(7) where

is a fixed non-negative integer with , and with .

, ,

Remark 1: In Step 1, the conditions (1) and (2) are to make sure that Steps 2 and 3 are available, respectively. By applying Construction A, more FHSs with different types of lengths can be obtained. Different from the constructions in [2], our construction generates an FHS set with the same number of sequences as the known FHS set but larger alphabet. Next, we study the alphabet of the FHS set . Lemma 3: For any integer with is one element of the sequence such that .

, if there , then

Then, for any with , by (6) we have

and with , i.e.,

.

Define a set (8) where denotes the times of occurring in . With the above preparations, we can determine the alphabet . Theorem 1: The alphabet

satisfies

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where for each

, and

is the unique pair of integers with such that is a

component of . Proof: For each , according to (5) of Step 3, there exists only one pair of integers with and such that . By Lemma 3, and then we have (9) For any integer

with

and

, there are one element of such that , where Lemma 3, we have

and

of

Proof: From Construction A, there are sequences in , each having length . By Theorem 1, the size of is . In what follows, for , we will analyze the Hamming correlation for two sequences , where . Let with and , and let with , and . Then, can be expressed as

, i.e., (13)

and one element

are two different elements of and . By

By (6) and (13), we have

(14) The fact implies . This is to say, (10)

By (9) and (10), we have (11)

where the addition in the subscript is performed modulo the length of its corresponding sequence. The discussion of can be divided into three cases according to and . Case I: and . By (1), (6), and (14), we have

On the other hand, by Lemma 4 to prove (12) it is sufficient to show that for any given , we have , where is an element of the sequence . Otherwise, indicates that there exist two non-negative integers and such that

where

, , and

, is an element of

, . Therefore, we

have

where the last identity holds due to for all elements , , . Since and is a primitive element of , for any given , runs through all elements of once as varies from 0 to . Note that equals 1 or 0. Further, it follows from Fact 1 that equals 1 if and

together with Fact 1.3 which leads to results in uniqueness of the pair tradiction. This shows that (12) holds. By (11) and (12), we have

. Then, the , a con.

Theorem 2: Assume that the FHS set has parameters . Then, the set in Construction A has parameters , where is the cardinality of the set defined by (8).

Therefore

, and 0 otherwise. That is,


Case II: and (1), (6), and (14) give

. The equalities

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Because of

, by Fact 1.2, we have

. Hence

Combining the aforementioned three cases, we finish the proof. IV. NEW OPTIMAL FHSS AND SETS OF FHSS In this section, by choosing suitable sets of FHSs, optimal FHSs and sets of FHSs with new parameters can be generated by Construction A. A. New Optimal FHSs By restricting in Construction A, some optimal FHSs with new parameters can be obtained. In this case, the expression (8) can be simplified as (15) where Note that

. . According to Fact 1

where denotes the times of occurring in the known FHS . In the case of , any FHS, in which each available frequency in exactly occurs once, is optimal with respect to the Lempel–Greenberger bound and has parameters . By Construction A and Theorem 2, the FHS constructed from this FHS has parameters

mod mod Therefore, if

mod

, we have where . So, it is also optimal. Hence, in the remainder of this section, we always assume that . The following result can be obtained by a direct application of Theorem 2 and Lemma 1.

mod mod Otherwise

mod

Proposition 2: Let be the constructed from an optimal A. Then, is optimal if

-FHS

-FHS by Construction

mod where Then, in the case of if

, we have mod and otherwise. Note that for any given and with and , there exists an integer with such that mod . This is to say that for each and with and , there exists an integer such that . Case III: and .

is the least nonnegative residue of modulo . The following proposition provides a convenient way to construct optimal and near-optimal FHSs.

Proposition 3: Let be the -FHS constructed from an optimal -FHS by Construction A. Then 1) if and , is also an optimal FHS with parameters , where with ; 2) if and , is near-optimal with parameters . 3) is balanced if and only if is balanced with . In this case, we have . Proof: 1) Since is an optimal FHS, by Proposition 1, we can assume

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and then

The facts

TABLE III SOME NEW OPTIMAL FHSS WITH PARAMETERS

and

imply

i.e., we have

. By , and then . By Proposition 1 and Theorem 2, is an optimal FHS with parameters . 2) Since is an optimal FHS and , by Proposition 1, we have for , and then

TABLE IV SOME NEW NEAR-OPTIMAL FHSS WITH PARAMETERS

(16)

Since , there exists an integer such that . Therefore, by Step 1 (2), we have . Since is an optimal FHS, by Proposition 1 and , we have . Note that means each available frequency exactly occurring in the FHS once. As a consequence, . This is to say implies . Then, the inequalities and give

By Proposition 1 and Theorem 2, we have that meets the Lempel–Greenberger bound. This shows that is a near-optimal FHS with parameters . 3) For a given frequency , by Step 3, there are exactly different elements of which are the components of the corresponding sequence , denoted by . By Lemma 3, for , if , then each element of appears in . By equality (6), if and , where , then there are two distinct integers , with , such that and . Thus, each element of appears times in and appears times in for . Hence, is balanced if and only if , for any , which is equivalent to . Hence, the assertion follows immediately. Table I in Section II lists some known optimal FHSs. Based on these sequences with parameters , by choosing suitable such that the conditions in Construction A and Proposition 3 are satisfied, some new optimal and near-optimal FHSs

can be obtained by applying Construction A for each integer with . For example, take , then

in the case of . Let be constructed from for by Construction A. By Proposition 3, some optimal FHSs are listed , and some in Table III with parameters near-optimal FHSs are listed in Table IV with parameters , where

can be obtained by carefully analyzing those constructions in [1]–[3], [5], [6], [10], [13], [14], [16]–[18], [22], and [25]. Fur, the value of is dependent on thermore, for the sequence the integers and [5], [6], [14], [25]. Specifically, in the case of , it can be verified that , and in the case and , there are some examples of of [5], [6], [14], [25]. For the sequence , if and only if or , and for the sequence , if and only if [1], [10], [13], [16], [18], [22], [25]. Compared the parameters with all known optimal and near-optimal FHSs, the sequences in Tables III and IV are new. By Proposition 3 (3), those new sequences in Table III are unbalanced.


If

, those sequences with parameters in Table III can be used to construct optimal FHSs again by Construction A and Proposition 3 (1). is satisThis is to say that if the condition fied, then Construction A can be applied to recursively construct more optimal FHSs. , , and be a primitive element generated by the primitive polynomial and . Then, is a primitive element of . Take

Example 1: Let of the finite field

,

for each integer with , we can obtain another with parameters based on optimal FHS . We do not list the sequence for brevity. These the FHS results are also verified by computer. B. New Optimal Sets of FHSs In this section, by restricting to be an optimal set of FHSs in Construction A, some optimal sets of FHSs with new parameters can also be obtained. By Theorem 2 and Lemma 2, we have the following. Proposition 4: Let be the set of FHSs constructed from by Conan optimal set of FHSs with parameters struction A. If

. In the case of

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and

(17) in Table I is an optimal FHS with parameters , , . Set for . Define a sequence and , where with is defined as follows: and there are integers in the set such if , then . Thus, that

By Construction A, take

, the FHS

then

is an optimal set of FHSs with parameters .

Remark 2: By the fact

is defined as if (18) then (17) holds. Since (18) is more convenient to verify, sometimes one can choose the parameters to satisfy (18) then they also satisfy (17). Table II in Section II gives some known optimal sets of FHSs. , by choosing Based on these sets with parameters suitable such that the conditions in both Construction A and Proposition 4 are satisfied, some optimal sets of FHSs can be obtained by applying Construction A for each integer with . Choose the parameters satisfying

where each non-negative integer in denotes the element , denotes the element 0, and is a primitive element of . It is easy to check that the FHS is optimal and has , which is consistent with Proposition 3. parameters By choosing with , , , , applying Construction A and Proposition 3 again,

as in Table V, then (18) holds. Let be constructed from for by Construction A, respectively. By Remark 2 and Proposition 4, those optimal sets of FHSs listed in Table V where is have parameters obtained by analyzing those constructions in [1], [2], [5]–[7], [14]–[18], [22], and [25]. Compared the parameters with all known optimal FHS sets, the FHS sets in Table V are new. in Note that those sets with parameters Table V are optimal. Therefore, they can also be used to construct other optimal sets of FHSs by Construction A. This is to say that for some suitable parameter and the parameters , if they satisfy the inequality (17), then based on these sets, our construction can be applied to construct optimal sets of FHSs recursively.

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and

TABLE V SOME NEW OPTIMAL SETS OF FHSS WITH PARAMETERS

Example 2: In the case of

,

,

, and

in Table II is an optimal set of FHSs with parameters . For , , , , and , the inequality (18) holds. With the same finite field as in Example 1, for every with , if , then define where appears times in . For each , if , we define , where altogether appears times in and . In this way, we have

and

where each non-negative integer in and denotes the eledenotes the element 0, and is a primitive element ment , . It is easy to check that the FHS set is optimal and of , which is consistent with Propohas parameters , the inequality (18) holds for sition 4. By choosing , , , and . In this parameters of FHSs with pacase, we construct another optimal set from for each integer with rameters . We do not list the sequence set for brevity. These results are also verified by computer. V. CONCLUSION

By Construction A and for , where

, we can obtain the FHS set

We have proposed an interleaving construction of FHS sets from known ones. Some optimal FHSs and sets of FHSs with new parameters are found. By recursively applying the proposed construction, one can obtain more optimal FHSs and sets of FHSs with new parameters. ACKNOWLEDGMENT The authors would like to thank the Associate Editor Professor Tor Helleseth and the two anonymous referees for their helpful comments, which have improved the presentation of this paper. REFERENCES [1] W. Chu and C. J. Colbourn, “Optimal frequency-hopping sequences via cyclotomy,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 1139–1141, Mar. 2005. [2] J. H. Chung, Y. K. Han, and K. Yang, “New classes of optimal frequency-hopping sequences by interleaving techniques,” IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5783–5791, Dec. 2009. [3] J. H. Chung and K. Yang, “Optimal frequency-hopping sequences with new parameters,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1685–1693, Apr. 2010. [4] J. H. Chung and K. Yang, “ -fold cyclotomy and its application to frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2306–2317, Apr. 2011.


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Xiangyong Zeng received the B.S. degree from the Department of Mathematics, Hubei University, Wuhan, China in 1995, and M.S. degree and Ph.D. degree from the Department of Mathematics, Beijing Normal University, Beijing, China in 1998 and 2002 respectively. From 2002 to 2004, he was a postdoctoral member in the Computer School of Wuhan University, Wuhan, China. He is currently a professor of Hubei University. His research interests include cryptography, sequence design and coding theory.

Han Cai received the B.S. degree in mathematics from Hubei University, Wuhan, China, in 2009. He is currently working toward the M.S. degree in Hubei University. His research interest includes sequence design.

Xiaohu Tang (M’04) received the B.S. degree in applied mathematics from the Northwest Polytechnic University, Xi’an, China, the M.S. degree in applied mathematics from the Sichuan University, Chengdu, China, and the Ph.D. degree in electronic engineering from the Southwest Jiaotong University, Chengdu, China, in 1992, 1995, and 2001 respectively. From 2003 to 2004, he was a postdoctoral member in the Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology. From 2007 to 2008, he was a visiting professor at the University of Ulm, Germany. Since 2001, he has been in the Institute of Mobile Communications, Southwest Jiaotong University, where he is currently a professor. His research interests include sequence design, coding theory and cryptography. Dr. Tang was the recipient of the National excellent Doctoral Dissertation award in 2003 (China), the Humboldt Research Fellowship in 2007 (Germany). He was the Guest Editor/Associate-Editor for special section on sequence design and its application in communications for IEICE Transactions Fundamentals.

Yang Yang received the B.S. and M.S. degrees in Hubei University, Wuhan, China, in 2005 and 2008, respectively. He is currently working toward the Ph.D. degree in Southwest Jiaotong University, Chengdu, China. His research interest includes sequences and cryptography.