ISRN Combinatorics

Volume 2013, Article ID 479408, 4 pages

http://dx.doi.org/10.1155/2013/479408

## Construction of Optimal Sets of Frequency Hopping Sequences

Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China

Received 18 December 2012; Accepted 3 January 2013

Academic Editors: C. da Fonseca and A. V. Kelarev

Copyright © 2013 Bin Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the set-theoretic characterization of an optimal set of FH sequences.

#### 1. Introduction

Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are an integral part of spread-spectrum communication systems such as FH-CDMA systems (for a description of such systems, see [1]). In modern radar and communication systems, frequency-hopping (FH) spread-spectrum techniques have become popular (see [2], for example).

Assume that is a set of available frequencies, called an *alphabet*. Let be the set of all sequences of length over . Any element of is called a *frequency hopping sequence* (FHS) of length over . Given two FH sequences, and , we define their Hamming correlation to be
where if and if , and where
and all operations among position indices are performed modulo . If , is the Hamming autocorrelation. If , is the Hamming cross correlation.

#### 2. Lower Bounds on the Correlations of FHSs

FH sequences for FH-CDMA systems are required to have good Hamming correlations and large linear span [3]; the linear span is defined to be the length of the shortest linear feedback shift register that can produce the sequence. FH sequences’ design normally involves six parameters: the size of the frequency library , the sequence length , the family size of the subset , the maximum out-of-phase Hamming autocorrelations , the maximum Hamming cross correlations , and the linear span. It is generally desired that the family of FH sequences has the following properties: (1)the maximum out-of-phase Hamming autocorrelations should be as small as possible,(2)the maximum Hamming cross correlations should be as small as possible,(3)the family size for given , , , and should be as large as possible,(4)the linear span should be as large as possible.

In order to evaluate the theoretical performance of the FH sequences, it is important to find some theoretical bounds for these parameters. Given , , and of , Lempel and Greenberger [4] and Peng and Fan [5] derived lower bounds on and of FH sequences in . We restate their results in this section, which will be used later as the criteria to determine whether the new FH sequences constructed in this paper are optimal or not.

For any single FH sequence , let be the maximum out-of-phase value of . If for all , that is, if the value is the least among all FH sequences of the same length and over the same frequency library , is called an optimal FH sequence. Lempel and Greenberger [4] developed the following lower bound for .

Lemma 1 (see [4]). *For every FH sequence of length over a frequency alphabet of size . One has
**
where is the least nonnegative residue of modulo .*

Corollary 2. *For every FH sequence of length over a frequency alphabet of size . One has
**
where with .*

For any given subset of containing FH sequences, any two , , we write . We define the maximum out-of-phase Hamming autocorrelations and the maximum Hamming cross correlations as

Peng and Fan [5] developed the following bounds on , which take into consideration the number of FH sequences in the set.

Lemma 3 (see [5]). *Let be a set of sequences of length over an alphabet of size . Define . Then
*

A family is an optimal set if the Peng-Fan lower bound (2) or (3) in Lemma 3 is met. Let , and we have the following corollary which is a useful tool to check the bound in Lemma 3.

Corollary 4 (see [6]). *Let be a set of sequences of length over an alphabet of size . Then
**
where with . *

A number of authors have made contributions to the construction of optimal FH sequences. Both algebraic and combinatorial constructions of optimal FH sequences have been given (see, for example, [3, 4, 7–11]). Most of them are concentrated on single optimal FH sequences. The purpose of this paper is to present a construction of optimal sets of FH sequences based on the set-theoretic characterization of an optimal set of FH sequences. Throughout what follows, we use -FHS to denote an FH sequence of length over an alphabet of size whose Hamming autocorrelation . We also call a set of FH sequences in a set of FH sequences, where .

#### 3. Mixed Difference Functions

Fuji-Hara et al. [9] characterized a -FHS in terms of partition-type cyclic difference packings. Given a partition of into subsets (*called base blocks*), we can define a difference function on given by . Let . Then is called a -PCDP (*partition-type cyclic difference packing*). Here, is used in the notation to indicate the number of base blocks, and is the list of the sizes of base blocks. This is to say that a -PCDP is a partition of into base blocks which satisfies the following property. For any fixed nonzero residue , the equation has at most solutions in the multiset union .

If we label the positions of a -FHS by the elements of , then, by the above definition, the sets of position indices of frequencies in form a -PCDP with for any nonzero . Conversely, if we label the base blocks of a -PCDP by the elements of and identify the frequency alphabet with , then the PCDP gives a -FHS in .

This fact reveals that a single FHS can be constructed by a PCDP. Apparently, the smaller the index of a PCDP the lower the Hamming autocorrelation of its corresponding FH sequence. For an optimal FH sequence, we need to construct a -PCDP so that its index is as small as possible for any given value of and . Based on Lempel-Greenberger bound on in Lemma 1, Fuji-Hara et al. [9] proved the following result.

Lemma 5 (see [9]). *There exists a -FHS over the alphabet if and only if there exists a -PCDP in . Furthermore, this FH sequence is optimal if for and if for .*

The correspondence between an individual FH sequence and a PCDP can be naturally extended to give a set-theoretic interpretation of a set of FH sequences. Let be a positive integer. Let be a collection of partitions of into subsets (*called base blocks*). Write , . For any ordered pair with , we define a difference function on given by For any integer with , we define a difference function on , as before, given by We refer to these difference functions defined above as *mixed difference functions* with respect to the given collection . Since each partition in determines uniquely an FH sequence, the collection gives a set of FH sequences in , and vice versa, where the alphabet is regarded as . For the optimality of the derived set of FH sequences from , we define , , , , , , . Then is a -PCDP, according to the previous definition. We say that is a collection of PCDPs. It turns out that there exists a set of FH sequences in if and only if there exists a collection of PCDPs in under our notations. This gives us an interpretation for a set of FH sequences from set-theoretic perspective. As with individual optimal FH sequence, for an optimal set of FH sequences, we are required to construct a collection of PCDPs so that its index is as small as possible. Since the index of is the same as the Hamming correlation , the Peng-Fan bounds in Lemma 3 can be employed as our benchmarks. As noted in [12], a set of FH sequences meeting one of the Peng-Fan bounds must be optimal. We have the following theorem.

Theorem 6 (see [13]). *Let be an integer. Then there exists a set of FH sequences in if and only if there exists a collection of PCDPs in . Furthermore, this set is optimal if meets one of the Peng-Fan lower bounds given in Lemma 3.*

#### 4. The Construction of Optimal Sets of FH Sequences

To begin with, we state some preparatory knowledge that will be used in our constructions.

Let be a power of prime , where is an arbitrary positive integer. Write for the absolute trace function from to defined by

We will use the following known result.

Lemma 7 (see [14, Corollary 7.17]). *For any the number of elements such that is . *

With the previous preparations, we are now ready to describe our construction of optimal sets of FH sequences.

Theorem 8. *Let be an odd prime and a positive integer. Define by , where is the trace function from to , is a generator of . Define
**
where all operations among the subscripts are performed modulo . Then is an optimal set of FH sequences over of length . Furthermore, each sequence of the set is optimal with respect to the bound of Lemma 1.*

*Proof. *For any with and any , we have
where the last equality follows from Lemma 7. Next we will show that for any with and any . For any with and any , we have
where the last equality follows from Lemma 7. It follows that is a collection of PCDPs in , where . By Theorem 6, it derives a set of FH sequences in . For this set, we have and both lower bounds in Lemma 3 are equal to by Corollary 4. Hence, it is optimal. Furthermore, each sequence of the set is optimal with respect to the bound of Lemma 1. The proof is then complete.

*Example 9. *Take and . It is readily calculated that
We then have

The family is a collection of 3 PCDPs in , where , , , and . By Theorem 6, it corresponds a set consisting of the following FH sequences over : with . It is optimal meeting the lower bounds in Lemma 3.

#### Acknowledgment

The Paper was supported by the Tian Yuan Special Funds of the National Natural Science Foundation of China (Grant no. 11226282).

#### References

- R. A. Scholtz, “Spread spectrum concept,”
*IEEE Transactions on Communications*, vol. 25, no. 8, pp. 748–755, 1977. View at Google Scholar · View at Scopus - P. Fan and M. Darnell,
*Sequence Design for Communications Applications*, Research Studies Press, Taunton, UK, 1996. - P. V. Kumar, “Frequency-hopping code sequence designs having large linear span,”
*IEEE Transactions on Information Theory*, vol. 34, no. 1, pp. 146–151, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - A. Lempel and H. Greenberger, “Families of sequences with optimal Hamming correlation properties,”
*IEEE Transactions on Information Theory*, vol. IT-20, pp. 90–94, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Peng and P. Fan, “Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,”
*IEEE Transactions on Information Theory*, vol. 50, no. 9, pp. 2149–2154, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - B. Wen, Y. Li, and C. Shi, “A construction of optimal sets of FH sequences,”
*The Australasian Journal of Combinatorics*, vol. 50, pp. 37–44, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Chu and C. J. Colbourn, “Optimal frequency-hopping sequences via cyclotomy,”
*IEEE Transactions on Information Theory*, vol. 51, no. 3, pp. 1139–1141, 2005. View at Publisher · View at Google Scholar · View at Scopus - C. Ding, M. J. Moisio, and J. Yuan, “Algebraic constructions of optimal frequency-hopping sequences,”
*IEEE Transactions on Information Theory*, vol. 53, no. 7, pp. 2606–2610, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Fuji-Hara, Y. Miao, and M. Mishima, “Optimal frequency hopping sequences: a combinatorial approach,”
*IEEE Transactions on Information Theory*, vol. 50, no. 10, pp. 2408–2420, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - G. Ge, R. Fuji-Hara, and Y. Miao, “Further combinatorial constructions for optimal frequency-hopping sequences,”
*Journal of Combinatorial Theory A*, vol. 113, no. 8, pp. 1699–1718, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Udaya and M. U. Siddiqi, “Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,”
*IEEE Transactions on Information Theory*, vol. 44, no. 4, pp. 1492–1503, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Ding and J. Yin, “Sets of optimal frequency-hopping sequences,”
*IEEE Transactions on Information Theory*, vol. 54, no. 8, pp. 3741–3745, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. Yin, “A construction of optimal sets of FH sequences,” in
*Coding and Cryptology*, vol. 4 of*Series on Coding Theory and Cryptology*, pp. 268–276, World Scientific, Hackensack, NJ, USA, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-X. Wan,
*Lectures on Finite Fields and Galois Rings*, World Scientific Publishing, River Edge, NJ, USA, 2003. View at MathSciNet