Optimal Frequency-Hopping Sequence Sets Based on Cyclotomy

In this paper, a kind of generalized cyclotomy with respect to the square of a prime is presented and the properties of the corresponding generalized cyclotomic numbers are investigated. Based on the generalized cyclotomy, a class of frequency-hopping sequence (FHS) set is constructed. By means of some basic properties of the generalized cyclotomy, we derive the Hamming correlation distribution of the new set. The results show that the proposed FHS set is optimal with regard to the average Hamming correlation (AHC) bound. By choosing suitable parameters, the construction also leads to the optimal FHS set and the optimal FHSs with regard to the maximum Hamming correlation (MHC) bound and Lempel-Greenberger bound, respectively.

[1]  Jin-Ho Chung,et al.  New Classes of Optimal Frequency-Hopping Sequences by Interleaving Techniques , 2009, IEEE Transactions on Information Theory.

[2]  A. Whiteman.  A family of difference sets , 1962 .

[3]  Xiaohu Tang,et al.  Optimal Frequency Hopping Sequences of Odd Length , 2013, IEEE Transactions on Information Theory.

[4]  Daiyuan Peng,et al.  Lower Bounds on the Hamming Auto- and Cross Correlations of Frequency-Hopping Sequences , 2004, IEEE Trans. Inf. Theory.

[5]  Abraham Lempel,et al.  Families of sequences with optimal Hamming-correlation properties , 1974, IEEE Trans. Inf. Theory.

[6]  Jin-Ho Chung,et al.  Optimal Frequency-Hopping Sequences With New Parameters , 2010, IEEE Transactions on Information Theory.

[7]  Daiyuan Peng,et al.  Generalised class of cubic frequency-hopping sequences with large family size , 2005 .

[8]  Charles J. Colbourn,et al.  Optimal frequency-hopping sequences via cyclotomy , 2005, IEEE Transactions on Information Theory.

[9]  Cunsheng Ding,et al.  Sets of Optimal Frequency-Hopping Sequences , 2008, IEEE Transactions on Information Theory.

[10]  T. Helleseth,et al.  New Generalized Cyclotomy and Its Applications , 1998 .

[11]  Daiyuan Peng,et al.  New Constructions for Optimal Sets of Frequency-Hopping Sequences , 2011, IEEE Transactions on Information Theory.

[12]  Lei Hu,et al.  New Optimal frequency Hopping sequence Sets from Balanced Nested difference Packings of Partition-Type , 2013, Int. J. Found. Comput. Sci..

[13]  P. Vijay Kumar,et al.  Frequency-hopping code sequence designs having large linear span , 1988, IEEE Trans. Inf. Theory.

[14]  Fang Liu,et al.  A new frequency-hopping sequence set based upon generalized cyclotomy , 2013, Des. Codes Cryptogr..

[15]  Jin-Ho Chung,et al.  $k$ -Fold Cyclotomy and Its Application to Frequency-Hopping Sequences , 2011, IEEE Transactions on Information Theory.

[16]  Zhengchun Zhou,et al.  New sets of frequency-hopping sequences with optimal Hamming correlation , 2014, Des. Codes Cryptogr..

[17]  X. H. Tang,et al.  The Average Hamming Correlation for the Cubic Polynomial Hopping Sequences , 2008, 2008 International Wireless Communications and Mobile Computing Conference.

[18]  Cunsheng Ding,et al.  Algebraic Constructions of Optimal Frequency-Hopping Sequences , 2007, IEEE Transactions on Information Theory.

[19]  Masakazu Jimbo,et al.  Sets of Frequency Hopping Sequences: Bounds and Optimal Constructions , 2009, IEEE Transactions on Information Theory.

[20]  Cunsheng Ding,et al.  Optimal Sets of Frequency Hopping Sequences From Linear Cyclic Codes , 2010, IEEE Transactions on Information Theory.

[21]  Gennian Ge,et al.  Optimal Frequency Hopping Sequences: Auto- and Cross-Correlation Properties , 2009, IEEE Trans. Inf. Theory.

[22]  Jin-Ho Chung,et al.  A New Class of Balanced Near-Perfect Nonlinear Mappings and Its Application to Sequence Design , 2013, IEEE Transactions on Information Theory.