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Three classes of partitioned difference families and their optimal constant composition codes
A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets
School of Mathematics and Big Data, Dezhou University, Dezhou, 253023, China |
In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.
References:
[1] |
Z. F. Cao, G. N. Ge and Y. Miao,
Combinatorial characterizations of one-coincidence frequency-hopping sequences, Des. Codes Cryptogr., 41 (2006), 177-184.
doi: 10.1007/s10623-006-9007-8. |
[2] |
W. D. Chen,
Frequency hopping patterns with wide intervals, J. Systems Sci. Math. Sci., 3 (1983), 295-303.
|
[3] |
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. |
[4] |
J. H. Chung and K. Yang,
Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inform. Theory, 56 (2010), 1685-1693.
doi: 10.1109/TIT.2010.2040888. |
[5] |
C. S. Ding and J. Yin,
Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.
doi: 10.1109/TIT.2008.926410. |
[6] |
C. S. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima,
Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304.
doi: 10.1109/TIT.2009.2021366. |
[7] |
C. S. Ding, M. 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. |
[8] |
R. Fuji-Hara, Y. 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. |
[9] |
L. Fukshansky and A. A. Shaar,
A new family of one-coincidence sets of sequences with dispersed elements for frequency-hopping CDMA systems, Adv. Math. Commun., 12 (2018), 181-188.
doi: 10.3934/amc.2018012. |
[10] |
G. N. Ge, Y. Miao and Z. H. 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] |
G. N. Ge, R. Fuji-Hara and Y. Miao,
Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718.
doi: 10.1016/j.jcta.2006.03.019. |
[12] |
L. Guan, Z. Li, J. B. Si and R. Gao,
Generation and characteristics analysis of cognitive-based high-performance wide-gap FH sequences, IEEE Trans. Veh. Technol., 64 (2015), 5056-5069.
doi: 10.1109/TVT.2014.2377299. |
[13] |
L. Guan, Z. Li, J. B. Si and Y. C. Huang, Generation and characterization of orthogonal FH sequences for the congnitive network, Sci. China Inf. Sci., 58 (2015), 1-11. Google Scholar |
[14] |
W. M. He and G. Feng, Comparison of two algorithms to generate wide gap FH code sequence, J. PLA Univ. Sci. Technol. (Nat. Sci.), 5 (2004), 29-33. Google Scholar |
[15] |
P. V. Kumar,
Frequency-hopping code sequence designs having large linear span, IEEE Trans. Inform. Theory, 34 (1988), 146-151.
doi: 10.1109/18.2616. |
[16] |
T. H. Lee, H. H. Jung and J. H. Chung, A new one-coincidence frquency-hopping sequene set of length $p^2-p$, Proc. ITW, (2018). Google Scholar |
[17] |
A. Lempel and H. Greenberger,
Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94.
doi: 10.1109/tit.1974.1055169. |
[18] |
B. Li, One-coincidence sequences with specified distance between adjacent symbols for frequency-hopping multiple access, IEEE Trans. Commun., 45 (1997), 408-410. Google Scholar |
[19] |
P. H. Li, C. L. Fan, Y. Yang and Y. Wang, New bounds on wide-gap frequency-hopping sequences, IEEE Commun. Letter, 23 (2019), 1050-1053. Google Scholar |
[20] |
X. H. Niu and C. P. Xing,
New extension construction of optimal frequency hopping sequence sets, IEEE Trans. Inform. Theory, 65 (2019), 5846-5855.
doi: 10.1109/TIT.2019.2916362. |
[21] |
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. |
[22] |
W. L. Ren, F. W. Fu and Z. C. Zhou,
New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Crypt., 72 (2014), 423-434.
doi: 10.1007/s10623-012-9774-3. |
[23] |
W. L. Ren, F. W. Fu, F. Wang and J. Gao, A class of optimal one-coincidence frequency-hopping sequence sets with composite length, IEICE Trans. Fund. Elec., Commun. Compu. Sci., E100-A (2017), 2528-2533. Google Scholar |
[24] |
A. A. Shaar and P. A. Davies,
A survey of one-coincidence sequences for frequency-hopped spread-spectrum systems, Proc. IEE-F, 131 (1984), 719-724.
|
[25] |
A. A. Shaar and P. A. Davies, Prime sequences: Quasi-optimal sequences for OR channel code division multiplexing, Electronics Letters, 19 (1983), 888-890. Google Scholar |
[26] |
M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, Inc., New York, 2002. Google Scholar |
[27] |
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. |
[28] |
H. Wang, Y. Zhao, F. Shen and W. Sun, The design of wide interval FH sequences based on RS code, Appl. Sci. Technol., 37 (2010), 28-33. Google Scholar |
[29] |
Y. Yang, X. H. Tang, U. Parampalli and D. Y. Peng,
New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inform. Theory, 57 (2011), 7605-7613.
doi: 10.1109/TIT.2011.2162571. |
[30] |
H. Q. Zhang, Design and performance analysis of frequency hopping sequences with given minimum gap, Proc. ICMMT (2010). Google Scholar |
show all references
References:
[1] |
Z. F. Cao, G. N. Ge and Y. Miao,
Combinatorial characterizations of one-coincidence frequency-hopping sequences, Des. Codes Cryptogr., 41 (2006), 177-184.
doi: 10.1007/s10623-006-9007-8. |
[2] |
W. D. Chen,
Frequency hopping patterns with wide intervals, J. Systems Sci. Math. Sci., 3 (1983), 295-303.
|
[3] |
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. |
[4] |
J. H. Chung and K. Yang,
Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inform. Theory, 56 (2010), 1685-1693.
doi: 10.1109/TIT.2010.2040888. |
[5] |
C. S. Ding and J. Yin,
Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.
doi: 10.1109/TIT.2008.926410. |
[6] |
C. S. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima,
Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304.
doi: 10.1109/TIT.2009.2021366. |
[7] |
C. S. Ding, M. 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. |
[8] |
R. Fuji-Hara, Y. 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. |
[9] |
L. Fukshansky and A. A. Shaar,
A new family of one-coincidence sets of sequences with dispersed elements for frequency-hopping CDMA systems, Adv. Math. Commun., 12 (2018), 181-188.
doi: 10.3934/amc.2018012. |
[10] |
G. N. Ge, Y. Miao and Z. H. 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] |
G. N. Ge, R. Fuji-Hara and Y. Miao,
Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718.
doi: 10.1016/j.jcta.2006.03.019. |
[12] |
L. Guan, Z. Li, J. B. Si and R. Gao,
Generation and characteristics analysis of cognitive-based high-performance wide-gap FH sequences, IEEE Trans. Veh. Technol., 64 (2015), 5056-5069.
doi: 10.1109/TVT.2014.2377299. |
[13] |
L. Guan, Z. Li, J. B. Si and Y. C. Huang, Generation and characterization of orthogonal FH sequences for the congnitive network, Sci. China Inf. Sci., 58 (2015), 1-11. Google Scholar |
[14] |
W. M. He and G. Feng, Comparison of two algorithms to generate wide gap FH code sequence, J. PLA Univ. Sci. Technol. (Nat. Sci.), 5 (2004), 29-33. Google Scholar |
[15] |
P. V. Kumar,
Frequency-hopping code sequence designs having large linear span, IEEE Trans. Inform. Theory, 34 (1988), 146-151.
doi: 10.1109/18.2616. |
[16] |
T. H. Lee, H. H. Jung and J. H. Chung, A new one-coincidence frquency-hopping sequene set of length $p^2-p$, Proc. ITW, (2018). Google Scholar |
[17] |
A. Lempel and H. Greenberger,
Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94.
doi: 10.1109/tit.1974.1055169. |
[18] |
B. Li, One-coincidence sequences with specified distance between adjacent symbols for frequency-hopping multiple access, IEEE Trans. Commun., 45 (1997), 408-410. Google Scholar |
[19] |
P. H. Li, C. L. Fan, Y. Yang and Y. Wang, New bounds on wide-gap frequency-hopping sequences, IEEE Commun. Letter, 23 (2019), 1050-1053. Google Scholar |
[20] |
X. H. Niu and C. P. Xing,
New extension construction of optimal frequency hopping sequence sets, IEEE Trans. Inform. Theory, 65 (2019), 5846-5855.
doi: 10.1109/TIT.2019.2916362. |
[21] |
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. |
[22] |
W. L. Ren, F. W. Fu and Z. C. Zhou,
New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Crypt., 72 (2014), 423-434.
doi: 10.1007/s10623-012-9774-3. |
[23] |
W. L. Ren, F. W. Fu, F. Wang and J. Gao, A class of optimal one-coincidence frequency-hopping sequence sets with composite length, IEICE Trans. Fund. Elec., Commun. Compu. Sci., E100-A (2017), 2528-2533. Google Scholar |
[24] |
A. A. Shaar and P. A. Davies,
A survey of one-coincidence sequences for frequency-hopped spread-spectrum systems, Proc. IEE-F, 131 (1984), 719-724.
|
[25] |
A. A. Shaar and P. A. Davies, Prime sequences: Quasi-optimal sequences for OR channel code division multiplexing, Electronics Letters, 19 (1983), 888-890. Google Scholar |
[26] |
M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, Inc., New York, 2002. Google Scholar |
[27] |
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. |
[28] |
H. Wang, Y. Zhao, F. Shen and W. Sun, The design of wide interval FH sequences based on RS code, Appl. Sci. Technol., 37 (2010), 28-33. Google Scholar |
[29] |
Y. Yang, X. H. Tang, U. Parampalli and D. Y. Peng,
New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inform. Theory, 57 (2011), 7605-7613.
doi: 10.1109/TIT.2011.2162571. |
[30] |
H. Q. Zhang, Design and performance analysis of frequency hopping sequences with given minimum gap, Proc. ICMMT (2010). Google Scholar |
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