IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 7, JULY 2012 4899 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 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), 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 mul- tiple-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 sequence of length from sequences of length . They have been widely used in constructing sequences with good periodic correlation [11], [12]. In 2010, Chung et al. intro- duced 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 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, Southwest Jiaotong 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 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 construc- tion 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) where if , and 0 otherwise, and the addi- 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