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doi: 10.3934/amc.2020131
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A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets

Wenli Ren , and  Feng Wang

School of Mathematics and Big Data, Dezhou University, Dezhou, 253023, China

* Corresponding author: Wenli Ren

Received  July 2020 Revised  November 2020 Early access January 2021

Fund Project: This research is supported in part by the Natural Science Foundation of Shandong Province (Grant Nos. ZR2018LA001, ZR2018LA003) and in part by the Program of Science and Technology of Shandong Province (Grant Nos. J16LI58, J18KB099)

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.

Keywords: Frequency-hopping sequence, Hamming correlation, one-coincidence.
Mathematics Subject Classification: Primary: 94A55; Secondary: 94A05.
Citation: Wenli Ren, Feng Wang. A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2020131
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.  Google Scholar

[2]

W. D. Chen, Frequency hopping patterns with wide intervals, J. Systems Sci. Math. Sci., 3 (1983), 295-303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[2]

W. D. Chen, Frequency hopping patterns with wide intervals, J. Systems Sci. Math. Sci., 3 (1983), 295-303.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H. Q. Zhang, Design and performance analysis of frequency hopping sequences with given minimum gap, Proc. ICMMT (2010). Google Scholar

Table 1.  The optimal OC-FHS set $ \mathcal{B}^w $ with FHS distances $ d(B_k^w) $
$ \mathcal{B}^w $ Frequencies $ d(B_k^w) $
$ B_1^7 $ $ (21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31) $ 7
$ B_2^7 $ $ (42, 56, 70, 67, 64, 61, 58, 55, 52, 49, 63, 60, 57, 54, 51, 48, 45) $ 3
$ B_3^7 $ $ (46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59) $ 4
$ B_4^7 $ $ (50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56) $ 6
$ B_5^7 $ $ (54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 48, 49, 50, 51, 52, 53) $ 1
$ B_6^7 $ $ (41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50) $ 8
$ B_7^7 $ $ (45, 60, 58, 56, 54, 52, 67, 65, 63, 61, 59, 57, 55, 53, 51, 49, 47) $ 2
$ B_8^7 $ $ (66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61) $ 5
$ B_9^7 $ $ (36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41) $ 5
$ B_{10}^{7} $ $ (57, 59, 61, 63, 65, 67, 52, 54, 56, 58, 60, 45, 47, 49, 51, 53, 55) $ 2
$ B_{11}^7 $ $ (61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52) $ 8
$ B_{12}^7 $ $ (48, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49) $ 1
$ B_{13}^7 $ $ (52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46) $ 6
$ B_{14}^7 $ $ (56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43) $ 4
$ B_{15}^7 $ $ (60, 63, 49, 52, 55, 58, 61, 64, 67, 70, 56, 42, 45, 48, 51, 54, 57) $ 3
$ B_{16}^7 $ $ (81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71) $ 7
Table Options

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$ \mathcal{B}^w $ Frequencies $ d(B_k^w) $
$ B_1^7 $ $ (21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31) $ 7
$ B_2^7 $ $ (42, 56, 70, 67, 64, 61, 58, 55, 52, 49, 63, 60, 57, 54, 51, 48, 45) $ 3
$ B_3^7 $ $ (46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59) $ 4
$ B_4^7 $ $ (50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56) $ 6
$ B_5^7 $ $ (54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 48, 49, 50, 51, 52, 53) $ 1
$ B_6^7 $ $ (41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50) $ 8
$ B_7^7 $ $ (45, 60, 58, 56, 54, 52, 67, 65, 63, 61, 59, 57, 55, 53, 51, 49, 47) $ 2
$ B_8^7 $ $ (66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61) $ 5
$ B_9^7 $ $ (36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41) $ 5
$ B_{10}^{7} $ $ (57, 59, 61, 63, 65, 67, 52, 54, 56, 58, 60, 45, 47, 49, 51, 53, 55) $ 2
$ B_{11}^7 $ $ (61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52) $ 8
$ B_{12}^7 $ $ (48, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49) $ 1
$ B_{13}^7 $ $ (52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46) $ 6
$ B_{14}^7 $ $ (56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43) $ 4
$ B_{15}^7 $ $ (60, 63, 49, 52, 55, 58, 61, 64, 67, 70, 56, 42, 45, 48, 51, 54, 57) $ 3
$ B_{16}^7 $ $ (81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71) $ 7
Table 2.  The optimal WG-OC-FHS set $ \mathcal{C}^7 $ with the FHS gap $ D = 3 $
$ \mathcal{C}^7 $ Frequencies $ d(B_k^w) $
$ B_1^7 $ $ (21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31) $ 7
$ B_3^7 $ $ (46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59) $ 4
$ B_4^7 $ $ (50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56) $ 6
$ B_6^7 $ $ (41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50) $ 8
$ B_8^7 $ $ (66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61) $ 5
$ B_9^7 $ $ (36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41) $ 5
$ B_{11}^7 $ $ (61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52) $ 8
$ B_{13}^7 $ $ (52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46) $ 6
$ B_{14}^7 $ $ (56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43) $ 4
$ B_{16}^7 $ $ (81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71) $ 7
Table Options

Download as excel

$ \mathcal{C}^7 $ Frequencies $ d(B_k^w) $
$ B_1^7 $ $ (21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31) $ 7
$ B_3^7 $ $ (46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59) $ 4
$ B_4^7 $ $ (50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56) $ 6
$ B_6^7 $ $ (41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50) $ 8
$ B_8^7 $ $ (66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61) $ 5
$ B_9^7 $ $ (36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41) $ 5
$ B_{11}^7 $ $ (61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52) $ 8
$ B_{13}^7 $ $ (52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46) $ 6
$ B_{14}^7 $ $ (56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43) $ 4
$ B_{16}^7 $ $ (81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71) $ 7
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