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Two classes of near-optimal frequency-hopping sequence sets with prime-power period

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Abstract

In this paper, a kind of generalized cyclotomy with respect to a prime power is introduced and properties of the corresponding generalized cyclotomic numbers are investigated. Based on the generalized cyclotomy, two classes of frequency-hopping sequence (FHS) sets with prime-power period are presented. Meanwhile, we derive the Hamming correlation distribution of the new FHS sets. The results show that the proposed FHSs and FHS sets have (near-) optimal maximum Hamming correlation (MHC). These classes of near-optimal FHS sets have new parameters which are not covered in the literature.

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References

  1. Lempel, A., Greenberger, H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20, 90-94 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Peng, D., Fan, P.: Lower bounds on the Hamming auto-and cross-correlations of frequency-hopping sequences. IEEE Trans. Inf. Theory 50, 2149-2154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Peng, D., Niu, X., Tang, X., Chen, Q.: The Average Hamming Correlation for the Cubic Polynomial Hopping Sequences International Wireless Communications and Mobile Computing Conference(IWCMC 2008), Crete Island, Greece, pp. 464-469 (2008)

  4. Ding, C., Fuji-Hara, R., Fujiwara, Y., Jimbo, M., Mishima, M.: Sets of frequency hopping sequences: bounds and optimal constructions. IEEE Trans. Inf. Theory 55, 3297-3304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kumar, P.: Frequency-hopping code sequence designs having large linear span. IEEE Trans. Inf. Theory 34, 146-151 (1988)

    Article  MathSciNet  Google Scholar 

  6. Fuji-Hara, R., Miao, Y., Mishima, M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inf. Theory 50, 2408-2420 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chu, W., Colbourn, C.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51, 1139-1141 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, C., Moisio, M., Yuan, J.: Algebraic constructions of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 53, 2606-2610 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, C., Yin, J.: Sets of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 54, 3741-3745 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ge, G., Miao, Y., Yao, Z.: Optimal frequency hopping sequences: auto-and cross-correlation properties. IEEE Trans. Inf. Theory 55, 867-879 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Han, Y., Yang, K.: On the Sidel’nikov sequences as frequency-hopping sequences. IEEE Trans. Inf. Theory 55, 4279-4285 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chung, J., Han, Y., Yang, K.: New classes of optimal frequency-hopping sequences by interleaving techniques. IEEE Trans. Inf. Theory 55, 5783-5791 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ding, C., Yang, Y., Tang, X.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 56, 3605-3612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung, J., Yang, K.: Optimal frequency-hopping sequences with new parameters. IEEE Trans. Inf. Theory 56, 1685-1693 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chung, J., Yang, K.: k-fold cyclotomy and its application to frequency-hopping Sequences. IEEE Trans. Inf. Theory 57, 2306-2317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhou, Z., Tang, X., Peng, D., Udaya, P.: New constructions for optimal sets of frequency-hopping sequences. IEEE Trans. Inf. Theory 57, 3831-3840 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zeng, X., Cai, H., Tang, X., Yang, Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59, 3237-3248 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Su, M.: New Optimum Frequency Hopping Sequences Derived from Fermat Quotients Sixth International Workshop on Signal Design and Its Applications in Communications, Tokyo, Japan, pp. 166-169 (2013)

    Chapter  Google Scholar 

  19. Chung, J., Yang K.: A new class of balanced near-perfect nonlinear mappings and its application to sequence design. IEEE Trans. Inf. Theory 59, 1090-1097 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, F., Peng, D., Zhou, Z., Tang, X.: New constructions of optimal frequency hopping sequences with new parameters. Adv. Math. Commun. 7, 91-101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ren, W., Fu, F., Zhou, Z.: New sets of frequency-hopping sequences with optimal Hamming correlation. Des. Codes Cryptogr. 72, 423-434 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, S., Cao, X., Xu, G.: Recursive construction of optimal frequency-hopping sequence sets. IET Commun. 10, 1080-1086 (2016)

    Article  Google Scholar 

  23. Xu, S., Cao, X., Xu, G.: Optimal frequency-hopping sequence sets based on cyclotomy. Int. J. Found. Comput. S. 27, 443-462 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, B., Lin, L., Ling S., Liu, H.: Three new classes of optimal frequency-hopping sequence sets. Des. Codes Cryptogr. 83, 219-232 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peng, D., Peng, T., Fan, P.: Generalized class of cubic frequency-hopping sequences with large family size. IEE Proc. Commun. 152, 897-902 (2005)

    Article  Google Scholar 

  26. Peng, D., Peng, T., Tang, X., Niu, X.: A Class of Optimal Frequency Hopping Sequences Based upon the Theory of Power Residues Sequences and Their Applications (SETA 2008), Lexington, KY, USA, September 14-18, pp. 188-196 (2008)

  27. Peng, D., Niu, X., Tang, X.: Average Hamming correlation for the cubic polynomial hopping sequences. IET Commun. 4, 1775-1786 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, F., Peng, D.: A class of frequency-hopping sequence family with optimal average Hamming correlation property. J. Electron. & Inf. Technol. 32, 1257-1261 (2010)

    Google Scholar 

  29. Chung, J., Yang, K.: New frequency-hopping sequence sets with optimal average and good maximum Hamming correlations. IET Commun. 6, 2048-2053 (2012)

    Article  Google Scholar 

  30. Ke, P., Zhang, H., Zhang, S.: New class of frequency-hopping sequence set with optimal average Hamming correlation property. J. Commun. 33, 168-175 (2012)

    Google Scholar 

  31. Liu, F., Peng, D., Zhou, Z.: A new frequency-hopping sequence set based upon generalized cyclotomy. Des. Codes Cryptogr. 69, 247-259 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, A., Zhou, Z., Feng, K.: A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Adv. Math. Commun. 9, 55-62 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gauss, C.: Disquisitiones Arithmeticae, 1801; English translation: Yale University Press, New Haven, 1966; Reprinted by Springer-Verlag, Berlin/Heidelberg/New York (1986)

  34. Whiteman, A.: A family of difference sets. Ill. J. Math. 6, 107-121 (1962)

    MathSciNet  MATH  Google Scholar 

  35. Ding, C., Helleseth, T.: New generalized cyclotomy and its applications. Finite Fields Appl. 4, 140-166 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fan, C., Ge, G.: A unified approach to Whiteman’s and Ding-Helleseth’s generalized cyclotomy over residue class rings. IEEE Trans. Inf. Theory 60, 1326-1336 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Apostol, T.: Introduction to analytic number theory. Springer-Verlag, NewYork (1976)

    MATH  Google Scholar 

  38. Park, K., Song, H., Kim, D., Golomb, S.: Optimal families of perfect polyphase sequences from the array structure of Fermat-Quotient sequences. IEEE Trans. Inf. Theory 62, 1076-1086 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11371011, 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) and the Nature Science Foundation of Jiangsu Province (Grant No. BK20160771).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Shanding Xu, Xiwang Cao, Guangkui Xu & Gaojun Luo

  2. Department of Mathematics and physics, Nanjing Institute of Technology, Nanjing, 211167, China

    Shanding Xu

  3. Key Laboratory of Mathematics and Interdisciplinary Sciences, Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China

    Shanding Xu

  4. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Xiwang Cao

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  1. Shanding Xu
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  2. Xiwang Cao
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  3. Guangkui Xu
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  4. Gaojun Luo
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Correspondence to Xiwang Cao.

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Xu, S., Cao, X., Xu, G. et al. Two classes of near-optimal frequency-hopping sequence sets with prime-power period. Cryptogr. Commun. 10, 437–454 (2018). https://doi.org/10.1007/s12095-017-0229-6

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  • DOI: https://doi.org/10.1007/s12095-017-0229-6

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