February  2013, 7(1): 91-101. doi: 10.3934/amc.2013.7.91

New constructions of optimal frequency hopping sequences with new parameters

1. 

The Thirtieth Research Institute, China Electronic Technology Group Corporation, Chengdu, China

2. 

School of Mobile Communications, Southwest Jiaotong University, Chengdu, 610031, China, China

3. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031

Received  June 2012 Published  January 2013

In this paper, three constructions of frequency hopping sequences (FHSs) are proposed using a new generalized cyclotomy with respect to $\textbf{Z}_{p^n}$, where $p$ is an odd prime and $n$ is a positive integer. Based on some basic properties of the new generalized cyclotomy, it is shown that all the constructed FHSs are optimal with respect to the well-known Lempel-Greenberger bound. Furthermore, these FHSs have new parameters which are not reported in the literature.
Citation: Fang Liu, Daiyuan Peng, Zhengchun Zhou, Xiaohu Tang. New constructions of optimal frequency hopping sequences with new parameters. Advances in Mathematics of Communications, 2013, 7 (1) : 91-101. doi: 10.3934/amc.2013.7.91
References:
[1]

T. M. Apostol, "Introduction to Analytic Number Theory,'', Springer-Verlag, (1976). Google Scholar

[2]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inform. Theory, 51 (2005), 1139. Google Scholar

[3]

J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inform. Theory, 56 (2010), 1685. Google Scholar

[4]

C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two,, IEEE Trans. Inform. Theory, 44 (1998), 1699. Google Scholar

[5]

C. Ding, R. Fuji-Hara, and Y. Fujiwara, Sets of frequency hopping sequences: bounds and optimal constructions,, IEEE Trans. Inform. Theory, 55 (2009), 3297. doi: 10.1109/TIT.2009.2021366. Google Scholar

[6]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications,, Finite Fields Appl., 4 (1998), 140. Google Scholar

[7]

C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_1^{e_1}\cdots p_t^{e_t}$,, IEEE Trans. Inform. Theory, 45 (1999), 467. Google Scholar

[8]

C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 53 (2007), 2606. doi: 10.1109/TIT.2007.899545. Google Scholar

[9]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 54 (2008), 3741. Google Scholar

[10]

P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency-hopping CDMA systems,, IEEE Trans. Inform. Theory, 4 (2005), 2836. Google Scholar

[11]

R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach,, IEEE Trans. Inform. Theory, 50 (2004), 2408. doi: 10.1109/TIT.2004.834783. Google Scholar

[12]

G. Ge, Y. Miao and Z. H. Yao, Optimal frequency hopping sequences: auto- and cross-correlation properties,, IEEE Trans. Inform. Theory, 55 (2008), 867. Google Scholar

[13]

Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences,, IEEE Trans. Inform. Theory, 55 (2009), 4279. Google Scholar

[14]

J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping,, IEEE J. Select. Areas Commun., 8 (1990), 819. Google Scholar

[15]

A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. doi: 10.1109/TIT.1974.1055169. Google Scholar

[16]

D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross-correlations of frequency-hopping sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2149. doi: 10.1109/TIT.2004.833362. Google Scholar

[17]

P. Udaya and M. N. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, IT-44 (1998), 1492. Google Scholar

[18]

A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107. Google Scholar

[19]

M. Z. Win and R. A. Scholtz, Ultra-Wide Bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,, IEEE Trans. Commun., 58 (2002), 679. Google Scholar

show all references

References:
[1]

T. M. Apostol, "Introduction to Analytic Number Theory,'', Springer-Verlag, (1976). Google Scholar

[2]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inform. Theory, 51 (2005), 1139. Google Scholar

[3]

J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inform. Theory, 56 (2010), 1685. Google Scholar

[4]

C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two,, IEEE Trans. Inform. Theory, 44 (1998), 1699. Google Scholar

[5]

C. Ding, R. Fuji-Hara, and Y. Fujiwara, Sets of frequency hopping sequences: bounds and optimal constructions,, IEEE Trans. Inform. Theory, 55 (2009), 3297. doi: 10.1109/TIT.2009.2021366. Google Scholar

[6]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications,, Finite Fields Appl., 4 (1998), 140. Google Scholar

[7]

C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_1^{e_1}\cdots p_t^{e_t}$,, IEEE Trans. Inform. Theory, 45 (1999), 467. Google Scholar

[8]

C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 53 (2007), 2606. doi: 10.1109/TIT.2007.899545. Google Scholar

[9]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 54 (2008), 3741. Google Scholar

[10]

P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency-hopping CDMA systems,, IEEE Trans. Inform. Theory, 4 (2005), 2836. Google Scholar

[11]

R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach,, IEEE Trans. Inform. Theory, 50 (2004), 2408. doi: 10.1109/TIT.2004.834783. Google Scholar

[12]

G. Ge, Y. Miao and Z. H. Yao, Optimal frequency hopping sequences: auto- and cross-correlation properties,, IEEE Trans. Inform. Theory, 55 (2008), 867. Google Scholar

[13]

Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences,, IEEE Trans. Inform. Theory, 55 (2009), 4279. Google Scholar

[14]

J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping,, IEEE J. Select. Areas Commun., 8 (1990), 819. Google Scholar

[15]

A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. doi: 10.1109/TIT.1974.1055169. Google Scholar

[16]

D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross-correlations of frequency-hopping sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2149. doi: 10.1109/TIT.2004.833362. Google Scholar

[17]

P. Udaya and M. N. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inform. Theory, IT-44 (1998), 1492. Google Scholar

[18]

A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107. Google Scholar

[19]

M. Z. Win and R. A. Scholtz, Ultra-Wide Bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,, IEEE Trans. Commun., 58 (2002), 679. Google Scholar

[1]

Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

[2]

Xing Liu, Daiyuan Peng. Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions. Advances in Mathematics of Communications, 2014, 8 (3) : 359-373. doi: 10.3934/amc.2014.8.359

[3]

Shanding Xu, Xiwang Cao, Jiafu Mi, Chunming Tang. More cyclotomic constructions of optimal frequency-hopping sequences. Advances in Mathematics of Communications, 2019, 13 (3) : 373-391. doi: 10.3934/amc.2019024

[4]

Xing Liu, Daiyuan Peng. Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques. Advances in Mathematics of Communications, 2017, 11 (1) : 151-159. doi: 10.3934/amc.2017009

[5]

Xianhua Niu, Daiyuan Peng, Zhengchun Zhou. New classes of optimal frequency hopping sequences with low hit zone. Advances in Mathematics of Communications, 2013, 7 (3) : 293-310. doi: 10.3934/amc.2013.7.293

[6]

Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004

[7]

Lenny Fukshansky, Ahmad A. Shaar. A new family of one-coincidence sets of sequences with dispersed elements for frequency hopping cdma systems. Advances in Mathematics of Communications, 2018, 12 (1) : 181-188. doi: 10.3934/amc.2018012

[8]

Pinhui Ke, Yueqin Jiang, Zhixiong Chen. On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation. Advances in Mathematics of Communications, 2018, 12 (3) : 525-539. doi: 10.3934/amc.2018031

[9]

Oǧuz Yayla. Nearly perfect sequences with arbitrary out-of-phase autocorrelation. Advances in Mathematics of Communications, 2016, 10 (2) : 401-411. doi: 10.3934/amc.2016014

[10]

Jingjun Bao. New families of strictly optimal frequency hopping sequence sets. Advances in Mathematics of Communications, 2018, 12 (2) : 387-413. doi: 10.3934/amc.2018024

[11]

Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006

[12]

Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209

[13]

Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041

[14]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015

[15]

Zilong Wang, Guang Gong, Rongquan Feng. A generalized construction of OFDM $M$-QAM sequences with low peak-to-average power ratio. Advances in Mathematics of Communications, 2009, 3 (4) : 421-428. doi: 10.3934/amc.2009.3.421

[16]

Tongjiang Yan, Yanyan Liu, Yuhua Sun. Cyclic codes from two-prime generalized cyclotomic sequences of order 6. Advances in Mathematics of Communications, 2016, 10 (4) : 707-723. doi: 10.3934/amc.2016036

[17]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

[18]

Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029

[19]

Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko, Serhiy Yanchuk. Frequency locking of modulated waves. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 847-875. doi: 10.3934/dcds.2011.31.847

[20]

Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]