Abstract
Direct-sequence spread spectrum and frequency-hopping (FH) spread spectrum are two main spread-coding technologies. Frequency-hopping sequences (FHSs) achieving the well-known Lempel–Greenberger bound play an important part in FH code-division multiple-access systems. Our objective is to construct more FHSs with new parameters attaining the above bound. In this paper, two classes of FHSs are proposed by means of two partitions of \({{\mathbb {Z}}_{v}}\), where v is an odd positive integer. It is shown that all the constructed FHSs are optimal with respect to the Lempel–Greenberger bound. By choosing appropriate injective functions, infinitely many optimal FHSs can be recursively obtained. Above all, these FHSs have new parameters which are not covered in the former literature.
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Lempel, A., Greenberger, H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20(1), 90–94 (1974)
Kumar, P.: Frequency-hopping code sequence designs having large linear span. IEEE Trans. Inf. Theory 34(1), 146–151 (1988)
Udaya, P., Siddiqi, M.: Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings. IEEE Trans. Inf. Theory 44(4), 1492–1503 (1998)
Peng, D., Fan, P.: Lower bounds on the Hamming auto- and cross- correlations of frequency-hopping sequences. IEEE Trans. Inf. Theory 50(9), 2149–2154 (2004)
Fuji-Hara, R., Miao, Y., Mishima, M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inf. Theory 50(10), 2408–2420 (2004)
Fan, P., Lee, M., Peng, D.: New family of hopping sequences for time/frequency-hopping CDMA systems. IEEE Trans. Wirel. Commun. 4(6), 2836–2842 (2005)
Chu, W., Colbourn, C.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51(3), 1139–1141 (2005)
Ge, G., Fuji-Hara, R., Miao, Y.: Further combinatorial constructions for optimal frequency-hopping sequences. J. Combinat. Theory A 113(8), 1699–1718 (2006)
Ding, C., Moisio, M., Yuan, J.: Algebraic constructions of optimal frequency hopping sequences. IEEE Trans. Inf. Theory 53(7), 2606–2610 (2007)
Ding, C., Yin, J.: Sets of optimal frequency hopping sequences. IEEE Trans. Inf. Theory 54(8), 3741–3745 (2008)
Ge, G., Miao, Y., Yao, Z.: Optimal frequency hopping sequences: auto- and cross-correlation properties. IEEE Trans. Inf. Theory 55(2), 867–879 (2009)
Han, Y., Yang, K.: On the Sidel’nikov sequences as frequency-hopping sequences. IEEE Trans. Inf. Theory 55(9), 4279–4285 (2009)
Chung, J., Yang, K.: Optimal frequency-hopping sequences with new parameters. IEEE Trans. Inf. Theory 56(4), 1685–1693 (2010)
Chung, J., Yang, K.: \(k\)-fold cyclotomy and its application to frequency-hopping sequences. IEEE Trans. Inf. Theory 57(4), 2306–2317 (2011)
Zeng, X., Cai, H., Tang, X., Yang, Y.: A class of optimal frequency hopping sequences with new parameters. IEEE Trans. Inf. Theory 58(7), 4899–4907 (2012)
Zeng, X., Cai, H., Tang, X., Yang, Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59(5), 3237–3248 (2013)
Cai, H., Zeng, X., Helleseth, T., Tang, X., Yang, Y.: A new construction of zero-difference balanced functions and its applications. IEEE Trans. Inf. Theory 59(8), 5008–5015 (2013)
Liu, F., Peng, D., Zhou, Z., Tang, X.: New constructions of optimal frequency hopping sequences with new parameters. Adv. Math. Commun. 7(1), 91–101 (2013)
Su, M.: New optimum frequency hopping sequences derived from fermat quotients. In: The Sixth International Workshop on Signal Design and Its Applications, Communications, pp. 166–169 (2013)
Cai, H., Zhou, Z., Yang, Y., Tang, X.: A new construction of frequency-hopping sequences with optimal partial Hamming correlation. IEEE Trans. Inf. Theory 60(9), 5782–5790 (2014)
Xu, S., Cao, X., Xu, G.: Recursive construction of optimal frequency-hopping sequence sets. IET Commun. 10(9), 1080–1086 (2016)
Cai, H., Zhou, Z., Tang, X., Miao, Y.: Zero-difference balanced functions with new parameters and their applications. IEEE Trans. Inf. Theory 63(7), 4379–4387 (2017)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Ding, C., Pei, D., Salomaa, A.: Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography. World Scientific, Singapore (1996)
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771007, 11601177 and 61572027). The first author was also supported by the Funding of Jiangsu Innovation Program for Graduate Education (Grant No. KYZZ15_0090), the Funding for Outstanding Doctoral Dissertation in NUAA (Grant No. BCXJ16-08), the Open Project Program of Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University (Grant No. GDSXJCKX2016-07), the Funding of Nanjing Institute of Technology (Grant No. CKJB201606), the Nature Science Foundation of Jiangsu Province (Grant No. BK20160771) and the Fundamental Research Funds for the Central Universities.
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Xu, S., Cao, X., Xu, G. et al. Two classes of optimal frequency-hopping sequences with new parameters. AAECC 30, 1–16 (2019). https://doi.org/10.1007/s00200-018-0356-0
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DOI: https://doi.org/10.1007/s00200-018-0356-0