February  2017, 11(1): 151-159. doi: 10.3934/amc.2017009

Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques

Provincial Key Laboratory of Information Coding and Transmission, Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

Received  July 2015 Published  February 2017

Fund Project: The authors are supported by the National Science Foundation of China (Grant No. 61271244), the Key Grant Project of Chinese Ministry of Education (Grant No. 311031 100), and the Young Innovative Research Team of Sichuan Province (Grant No. 2011JTD0007).

Aperiodic Hamming correlation is an important criterion for evaluating the goodness of frequency hopping (FH) sequence design, while it received little attraction in the literature. In this paper, a construction of FH sequences with optimal aperiodic Hamming correlation by interleaving techniques is presented. Further, a class of one-coincidence FH sequence sets under aperiodic Hamming correlation is proposed. By employing the one-coincidence FH sequence sets, a class of FH sequence sets with optimal aperiodic Hamming correlation is also constructed by interleaving techniques.

Citation: 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
References:
[1]

The Bluetooth Special Interest Group (SIG), Specification of the Bluetooth Systems-Core, available at http://www.bluetooth.com.

[2]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inform. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.

[3]

J. H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inform. Theory, 55 (2009), 5783-5791.  doi: 10.1109/TIT.2009.2032742.

[4]

C. DingM. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610.  doi: 10.1109/TIT.2007.899545.

[5]

C. DingY. Yang and X. H. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inform. Theory, 55 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.

[6]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.

[7]

Y. C. EunS. Y. JinY. P. Hong and H. Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inform. Theory, 50 (2004), 2438-2442.  doi: 10.1109/TIT.2004.834792.

[8] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996. 
[9]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach, IEEE Trans. Inform. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.

[10]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: auto-and cross-correlation properties, IEEE Trans. Inform. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.

[11] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar, Cambridge Univ. Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[12]

G. Gong and H. Y. Song, Two-tuple balance of non-binary sequences with ideal two-level autocorrelation, Discrete Appl. Math., 154 (2006), 2590-2598.  doi: 10.1016/j.dam.2006.04.025.

[13]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998,1767-1853.

[14]

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

[15]

X. Liu and D. Y. Peng, Sets of frequency hopping sequences under aperiodic Hamming correlation: upper bound and optimal constructions, Adv. Math. Commun., 8 (2014), 359-373.  doi: 10.3934/amc.2014.8.359.

[16]

X. LiuD. Y. Peng and H. Y. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Cryptogr., 73 (2014), 167-176.  doi: 10.1007/s10623-013-9817-4.

[17]

X. Liu, D. Y. Peng, X. H. Niu and F. Liu, Lower bounds on the aperiodic Hamming correlations of frequency hopping sequences, IEICE Trans. Fundam. Electr. Commun. Comp. Sci. , E96-A (2013), 1445-1450. doi: 10.1587/transfun.E96.A.1445.

[18]

X. H. NiuD. Y. Peng and Z. C. Zhou, Frequency/time hopping sequence sets with optimal partial Hamming correlation properties, Sci. China Ser. F Inf. Sci., 55 (2012), 2207-2215.  doi: 10.1007/s11432-012-4620-9.

[19]

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), 2149-2154.  doi: 10.1109/TIT.2004.833362.

[20]

D. Y. Peng, T. Peng, X. H. Tang and X. H. Niu, A class of optimal frequency hopping sequences based upon the theory of power residues, in Proc. 5th Int. Conf. Seq. Appl. , 2008,188-196. doi: 10.1007/978-3-540-85912-3_18.

[21]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inform. Theory, 44 (1998), 1492-1503.  doi: 10.1109/18.681324.

[22]

Z. C. ZhouX. H. TangX. H. Niu and P. Udaya, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inform. Theory, 58 (2012), 453-458.  doi: 10.1109/TIT.2011.2167126.

show all references

References:
[1]

The Bluetooth Special Interest Group (SIG), Specification of the Bluetooth Systems-Core, available at http://www.bluetooth.com.

[2]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inform. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.

[3]

J. H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inform. Theory, 55 (2009), 5783-5791.  doi: 10.1109/TIT.2009.2032742.

[4]

C. DingM. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610.  doi: 10.1109/TIT.2007.899545.

[5]

C. DingY. Yang and X. H. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inform. Theory, 55 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.

[6]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.

[7]

Y. C. EunS. Y. JinY. P. Hong and H. Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inform. Theory, 50 (2004), 2438-2442.  doi: 10.1109/TIT.2004.834792.

[8] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996. 
[9]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach, IEEE Trans. Inform. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.

[10]

G. GeY. Miao and Z. Yao, Optimal frequency hopping sequences: auto-and cross-correlation properties, IEEE Trans. Inform. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.

[11] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar, Cambridge Univ. Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[12]

G. Gong and H. Y. Song, Two-tuple balance of non-binary sequences with ideal two-level autocorrelation, Discrete Appl. Math., 154 (2006), 2590-2598.  doi: 10.1016/j.dam.2006.04.025.

[13]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Elsevier, Amsterdam, 1998,1767-1853.

[14]

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

[15]

X. Liu and D. Y. Peng, Sets of frequency hopping sequences under aperiodic Hamming correlation: upper bound and optimal constructions, Adv. Math. Commun., 8 (2014), 359-373.  doi: 10.3934/amc.2014.8.359.

[16]

X. LiuD. Y. Peng and H. Y. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Cryptogr., 73 (2014), 167-176.  doi: 10.1007/s10623-013-9817-4.

[17]

X. Liu, D. Y. Peng, X. H. Niu and F. Liu, Lower bounds on the aperiodic Hamming correlations of frequency hopping sequences, IEICE Trans. Fundam. Electr. Commun. Comp. Sci. , E96-A (2013), 1445-1450. doi: 10.1587/transfun.E96.A.1445.

[18]

X. H. NiuD. Y. Peng and Z. C. Zhou, Frequency/time hopping sequence sets with optimal partial Hamming correlation properties, Sci. China Ser. F Inf. Sci., 55 (2012), 2207-2215.  doi: 10.1007/s11432-012-4620-9.

[19]

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), 2149-2154.  doi: 10.1109/TIT.2004.833362.

[20]

D. Y. Peng, T. Peng, X. H. Tang and X. H. Niu, A class of optimal frequency hopping sequences based upon the theory of power residues, in Proc. 5th Int. Conf. Seq. Appl. , 2008,188-196. doi: 10.1007/978-3-540-85912-3_18.

[21]

P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inform. Theory, 44 (1998), 1492-1503.  doi: 10.1109/18.681324.

[22]

Z. C. ZhouX. H. TangX. H. Niu and P. Udaya, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inform. Theory, 58 (2012), 453-458.  doi: 10.1109/TIT.2011.2167126.

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