IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012 4899

A Class of Optimal Frequency HoppingSequences with New ParametersXiangyong Zeng, Han Cai, Xiaohu Tang, Member, IEEE, and Yang Yang

Abstract—In this paper, we propose an interleaving constructionof new sets of frequency hopping sequences from the known ones.By choosing suitable known optimal frequency hopping sequencesand sets of frequency hopping sequences and then recursively ap-plying the proposed construction, optimal frequency hopping se-quences and sets of frequency hopping sequences with new param-eters can be obtained.

Index Terms—Frequency hopping sequence (FHS), Hammingcorrelation, interleaving technique, Lempel–Greenberger bound,Peng–Fan bound.

I. INTRODUCTION

F REQUENCY hopping multiple-access is widely used inmodern communication systems such as ultrawideband,

military communications, Bluetooth, and so on [21]. In thosesystems, we have to minimize the maximum of Hammingout-of-phase autocorrelation and cross correlation of the setof frequency hopping sequences (FHSs) to reduce the mul-tiple-access interference. To accommodate many users, it is alsovery desirable that size of the FHS sets is as large as possible.However, the parameters of the FHS sets are subjected to sometheoretic 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 withrespect to the bounds. During the decades, numerous algebraicand combinatorial constructions of optimal FHSs and FHS setshave been proposed (see [1]–[10], [13]–[18], [22]–[25], andreferences therein).The interleaving technique is a method to construct a long

sequence of length from sequences of length . Theyhave been widely used in constructing sequences with goodperiodic correlation [11], [12]. In 2010, Chung et al. intro-duced interleaving technique to the design of FHSs with goodHamming correlation [2]. Based on known optimal FHS sets,they presented new classes of optimal FHSs with respect tothe Lempel–Greenberger bound and the Peng–Fan bound via

Manuscript received January 03, 2012; revised April 08, 2012; accepted April10, 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 ScienceFoundation of China (NSFC) under Grant 61170257. The work of X. Tang andY. Yang was supported by the NSFC under Grant 61171095.X. Zeng and H. Cai are with the Faculty of Mathematics and Computer Sci-

ence, Hubei University,Wuhan 430062, China (e-mail: xiangyongzeng@yahoo.com.cn; hancai_s@yahoo.cn).X. Tang and Y. Yang are with the Provincial Key Lab of Information

Coding and Transmission, Institute of Mobile Communications, SouthwestJiaotong University, Chengdu 610031, China (e-mail: xhutang@ieee.org;yang-data@yahoo.cn).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 optimalFHS set constructed by their method can be arranged into amatrix such that each column of the new FHS is exactly an FHSin the corresponding known FHS set. Compared to the originalone, the new set has longer sequence length, larger Hammingcorrelation, 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 meansof interleaving technique. We present a construction of FHSsets based on known ones, which results in new FHS setshaving a longer sequence length, the same maximum nontrivialHamming correlation, and the same number of sequences but alarger size of alphabet. Roughly speaking, in contrast to Chunget al.’s method, our interleaving approach improves the size ofsequence sets but with larger alphabet, if applied to the sameknown FHS sets. In our study, some known optimal FHSs(respectively, optimal sets of FHSs) are used in the proposedconstruction to construct new optimal FHSs (respectively,optimal sets of FHSs). We list the parameters of some optimalFHSs and sets of FHSs obtained by the proposed construc-tion in Section IV. Furthermore, if the parameters of thesesequences satisfy some extra conditions, then they can be usedto 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 FHSsets is proposed in Section III, and the properties of these FHSsets are also analyzed. In Section IV, some optimal FHSs andsets of FHSs are obtained based on known optimal FHSs andsets of FHSs, respectively. Section V concludes the study.

II. PRELIMINARIES

For a positive integer , let be a setof available frequencies, also called the alphabet. A sequence

is called an FHS of length over iffor all . For two FHSs and

of length over , their Hamming correlationis defined by

(1)

where if , and 0 otherwise, and the addi-tion in the subscript is performed modulo . If for all

, then we say and call theHamming autocorrelation of , denoted by for short.Define as

0018-9448/$31.00 © 2012 IEEE

4900 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012

for an FHS , and as

for two FHSs and such that .Throughout this paper, let denote an FHS of lengthover 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 iscalled balanced; otherwise, is called unbalanced. For a realnumber , let denote the smallest integer not less than and

denote the integer part of . A lower bound of wasestablished by Lempel and Greenberger as below.

Lemma 1 (see [18]): For every FHS of length over analphabet of size

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-optimalif , that is to say, is near-optimal withrespect 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 fol-lowing 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 of

size , and the maximum nontrivial Hamming correlationof the sequence set is defined by

We use to denote the set with . In thiscase, 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 thefollowing bounds on by taking account of the number ofsequences in the set .

Lemma 2 (see [20]): Let be a set of FHSs of lengthover an alphabet of size . Define . Then

(3)

and

(4)

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

III. A CONSTRUCTION OF FHS SETS

For a prime power , let denote the finite field withelements and be its extension field with degree , whereis a positive integer [19]. Let be a primitive element of the

finite field , i.e., . It is well knownthat there exist elements insuch 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 twoprimes (see [2], [3], and [10]), a factor of for a primeand 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 setover the alphabet with parameters

satisfying1) ;2) for each with ,where denotes the times of occurring in .

Step 2: Choose different integers fromthe set .Step 3: For each with , define an FHS

associated with the FHS ofin 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 ;3) if and only if and .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)

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TABLE ISOME KNOWN OPTIMAL FHSS WITH PARAMETERS

TABLE IISOME KNOWN OPTIMAL SETS OF FHSS WITH PARAMETERS

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

.

Remark 1: In Step 1, the conditions (1) and (2) are to makesure 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 constructionsin [2], our construction generates an FHS set with the samenumber of sequences as the known FHS set but largeralphabet.Next, we study the alphabet of the FHS set .

Lemma 3: For any integer with , if thereis one element of the sequence such that , then

.

Proof: Obviously for any integer with ,ranges over with varying from 0 to . By (6),

we then have

The conclusion immediately follows from Fact 1.1.

Lemma 4: The alphabet satisfies .

Proof: Given with and, straightforwardly

(7)

Then, for any with and with

, by (6) we have , i.e., .

Define a set

(8)

where denotes the times of occurring in . With theabove preparations, we can determine the alphabet .

Theorem 1: The alphabet satisfies

4902 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012

where for each , is the unique pair of integers withand such that is a

component of .Proof: For each , according to (5) of Step 3, there

exists only one pair of integers withand such that . By Lemma 3,

we have and then

(9)

For any integer with and , i.e.,

, there are one element of and one element

of such that and are two different elements of, where and . By

Lemma 3, we have

The fact implies. This is to say,

(10)

By (9) and (10), we have

(11)

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 integersand such that

where , , ,, and is an element of . Therefore, we

have

together with Fact 1.3 which leads to . Then, the

uniqueness of the pair results in , a con-tradiction. This shows that (12) holds.

By (11) and (12), we have .

Theorem 2: Assume that the FHS set has parameters. Then, the set in Construction A has parameters

, where is the cardinality of theset defined by (8).

Proof: From Construction A, there are sequences in ,each having length . By Theorem 1, the size of is

.In what follows, for , we will ana-

lyze the Hamming correlation for two sequences, where . Let

with and , and letwith , and .Then, can be expressed as

(13)

By (6) and (13), we have

(14)

where the addition in the subscript is performed modulothe length of its corresponding sequence. The discussion of

can be divided into three cases according toand .

Case I: and .By (1), (6), and (14), we have

where the last identity holds due tofor all elements , , .Since and is a primitive ele-ment of , for any given , runsthrough 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 , and 0 otherwise. That is,

Therefore

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Case II: and . The equalities(1), (6), and (14) give

where .Note that . According to Fact 1

modmod

Therefore, if mod , we have

modmod

Otherwise

mod

mod

Then, in the case of , we haveif mod and

otherwise. Note that forany given and with and

, there exists an integer withsuch that mod . Thisis to say that for each and withand , there exists an integer such that

.Case III: and .

Because of , by Fact 1.2, we have . Hence

Combining the aforementioned three cases, we finish theproof.

IV. NEW OPTIMAL FHSS AND SETS OF FHSS

In this section, by choosing suitable sets of FHSs, optimalFHSs and sets of FHSs with new parameters can be generatedby Construction A.

A. New Optimal FHSs

By restricting in Construction A, some optimal FHSswith new parameters can be obtained. In this case, the expres-sion (8) can be simplified as

(15)

where denotes the times of occurring in the known FHS.In the case of , any FHS, in which each available fre-

quency in exactly occurs once, is optimal with respect tothe Lempel–Greenberger bound and has parameters .By Construction A and Theorem 2, the FHS constructed fromthis FHS has parameters

where . So, it is also optimal. Hence, in the remainderof this section, we always assume that .The following result can be obtained by a direct application

of Theorem 2 and Lemma 1.

Proposition 2: Let be the -FHSconstructed from an optimal -FHS by ConstructionA. Then, is optimal if

where is the least nonnegative residue of modulo.

The following proposition provides a convenient way to con-struct optimal and near-optimal FHSs.

Proposition 3: Let be the -FHSconstructed from an optimal -FHS by ConstructionA. Then1) if and , is also an optimal FHSwith parameters , wherewith ;

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 as-sume

4904 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012

and then

The facts and imply

i.e., . By ,we have and then

. By Proposition 1 and Theorem 2,is an optimal FHSwith parameters .

2) Since is an optimal FHS and , by Proposition 1, wehave for , and then

(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 availablefrequency exactly occurring in the FHS once. As a con-sequence, . This is to say implies .Then, the inequalities and give

By Proposition 1 and Theorem 2, we have that meetsthe Lempel–Greenberger bound. This shows that is anear-optimal FHS with parameters .

3) For a given frequency , by Step 3, there are ex-actly different elements of which are thecomponents of the corresponding sequence , de-noted by . By Lemma 3, for

, if , then each elementof appears in . By equality(6), if and

, where, then there are two distinct

integers , with , suchthat and .Thus, each element of ap-pears times in and appears timesin for . Hence, is balanced ifand only if ,for any , which is equivalent to

. Hence, the assertion followsimmediately.

Table I in Section II lists some known optimal FHSs. Basedon these sequences with parameters , by choosing suit-able such that the conditions in Construction A and Proposi-tion 3 are satisfied, some new optimal and near-optimal FHSs

TABLE IIISOME NEW OPTIMAL FHSS WITH PARAMETERS

TABLE IVSOME NEW NEAR-OPTIMAL FHSS WITH PARAMETERS

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

in the case of .Let be constructed from for by Con-

struction A. By Proposition 3, some optimal FHSs are listedin Table III with parameters , and somenear-optimal FHSs are listed in Table IVwith parameters

, where

can be obtained by carefully analyzing those constructions in[1]–[3], [5], [6], [10], [13], [14], [16]–[18], [22], and [25]. Fur-thermore, for the sequence , the value of is dependent onthe integers and [5], [6], [14], [25]. Specifically, in the caseof , it can be verified that , and in the caseof and , there are some examples of[5], [6], [14], [25]. For the sequence , if and only if

or , and for the sequence , if andonly if [1], [10], [13], [16], [18], [22], [25]. Compared theparameters with all known optimal and near-optimal FHSs, thesequences in Tables III and IV are new. By Proposition 3 (3),those new sequences in Table III are unbalanced.

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If , those sequences with parametersin Table III can be used to construct

optimal FHSs again by Construction A and Proposition 3 (1).This is to say that if the condition is satis-fied, then Construction A can be applied to recursively constructmore optimal FHSs.

Example 1: Let , , and be a primitive elementof the finite field generated by the primitive polynomial

and . Then,is a primitive element of . Take

.In the case of , and

in Table I is an optimal FHS with parameters , ,and . Set for . Define a sequence

, where with is defined as follows:if and there are integers in the set suchthat , then . Thus,

By Construction A, take , the FHS is defined as

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

parameters , which is consistent with Proposition 3.By choosing with , ,, , applying Construction A and Proposition 3 again,

for each integer with , we can obtain anotheroptimal FHS with parameters based onthe FHS . We do not list the sequence for brevity. Theseresults are also verified by computer.

B. New Optimal Sets of FHSs

In this section, by restricting to be an optimal set of FHSs inConstructionA, some optimal sets of FHSswith new parameterscan also be obtained.By Theorem 2 and Lemma 2, we have the following.

Proposition 4: Let be the set of FHSs constructed froman optimal set of FHSs with parameters by Con-struction A. If

(17)

then is an optimal set of FHSs with parameters.

Remark 2: By the fact

if

(18)

then (17) holds. Since (18) is more convenient to verify, some-times one can choose the parameters to satisfy (18) then theyalso satisfy (17).Table II in Section II gives some known optimal sets of FHSs.

Based on these sets with parameters , by choosingsuitable such that the conditions in both Construction A andProposition 4 are satisfied, some optimal sets of FHSs can beobtained by applying Construction A for each integer with

.Choose the parameters satisfying

as in Table V, then (18) holds. Let be constructed from forby Construction A, respectively. By Remark 2

and Proposition 4, those optimal sets of FHSs listed in Table Vhave parameters where isobtained by analyzing those constructions in [1], [2], [5]–[7],[14]–[18], [22], and [25]. Compared the parameters with allknown optimal FHS sets, the FHS sets in Table V are new.Note that those sets with parameters in

Table V are optimal. Therefore, they can also be used to con-struct other optimal sets of FHSs by Construction A. This isto say that for some suitable parameter and the parameters

, if they satisfy the inequality (17), then basedon these sets, our construction can be applied to construct op-timal sets of FHSs recursively.

4906 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012

TABLE VSOME 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 whereappears times in . For each , if

, we define , where altogether appearstimes in and . In this way, we have

and

By Construction A and for , we can obtain the FHS set, where

and

where each non-negative integer in and denotes the ele-ment , denotes the element 0, and is a primitive elementof . It is easy to check that the FHS set is optimal andhas parameters , which is consistent with Propo-sition 4. By choosing , the inequality (18) holds forparameters , , , and . In thiscase, we construct another optimal set of FHSs with pa-rameters from for each integer with

. We do not list the sequence set for brevity.These results are also verified by computer.

V. CONCLUSION

We have proposed an interleaving construction of FHS setsfrom known ones. Some optimal FHSs and sets of FHSs withnew parameters are found. By recursively applying the proposedconstruction, one can obtain more optimal FHSs and sets ofFHSs with new parameters.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor Pro-fessor Tor Helleseth and the two anonymous referees for theirhelpful comments, which have improved the presentation of thispaper.

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Xiangyong Zeng received the B.S. degree from the Department of Mathe-matics, 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 apostdoctoral member in the Computer School of Wuhan University, Wuhan,China. He is currently a professor of Hubei University. His research interestsinclude 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 inHubei University. His research interest includes sequence design.

Xiaohu Tang (M’04) received the B.S. degree in applied mathematics fromthe Northwest Polytechnic University, Xi’an, China, the M.S. degree in appliedmathematics 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 Elec-

trical and Electronic Engineering, The Hong Kong University of Science andTechnology. From 2007 to 2008, he was a visiting professor at the Universityof Ulm, Germany. Since 2001, he has been in the Institute of Mobile Commu-nications, Southwest Jiaotong University, where he is currently a professor. Hisresearch 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).Hewas the Guest Editor/Associate-Editor for special section on sequence designand 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 interestincludes sequences and cryptography.