# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021019
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## On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $4p^n$

 Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China

* Corresponding author: Xiangyong Zeng

Received  January 2021 Revised  April 2021 Early access June 2021

Fund Project: The work was supported by Application foundation frontier project of Wuhan Science and Technology Bureau under Grant 2020010601012189, and by the National Nature Science Foundation of China (NSFC) under Grant 62072161. The work of Sun was supported by the Natural Science Foundation of Hubei under Grant 2019CFB544. The work of Zhang was supported by the Research Foundation of Education Bureau of Hubei Province, China under Grant D2020104

In this paper, a new class of generalized cyclotomic binary sequences with period $4p^n$ is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.

Citation: Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang. On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $4p^n$. Advances in Mathematics of Communications, doi: 10.3934/amc.2021019
##### References:
 [1] E. Bai, X. Liu and G. Xiao, Linear complexity of new generalized cyclotomic sequences of order two of length $pq$, IEEE Trans. Inf. Theory, 51 (2005), 1849-1853.  doi: 10.1109/TIT.2005.846450.  Google Scholar [2] D.M. Burton, Elementary Number Theory, McGram-Hill, New York, 1998.  Google Scholar [3] T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004.  Google Scholar [4] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order $2$, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.  Google Scholar [5] C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inf. Theory, 44 (1998), 1699-1702.  doi: 10.1109/18.681354.  Google Scholar [6] C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.  doi: 10.1006/ffta.1998.0207.  Google Scholar [7] C. Ding and T. Helleseth, Generalized cyclotomy codes of length $p^{m_{1}}_{1}p^{m_{2}}_{2}\cdots p^{m_{t}}_{t}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.  doi: 10.1109/18.748996.  Google Scholar [8] X. Dong, Linear complexity of generalized cyclotomic binary sequences of length $4p^n$, Inf. Sci. Lett., 4 (2015), 67-70.   Google Scholar [9] V. Edemskiy, About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$, Des. Codes Cryptogr., 61 (2011), 251-260.  doi: 10.1007/s10623-010-9474-9.  Google Scholar [10] V. Edemskiy and O. Antonova, The evaluation of the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2^np^m$, International Journal of Mathematical Models and Methods in Applied Sciences, 9 (2015), 512-517.   Google Scholar [11] V. Edemskiy, C. Li, X. Zeng and T. Helleseth, The linear complexity of generalized cyclotomic binary sequences of period $p^n$, Des. Codes Cryptogr., 87 (2018), 1183-1197.  doi: 10.1007/s10623-018-0513-2.  Google Scholar [12] V. Edemskiy and C. Wu, On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo $2^np^m$, WSEAS Transactions on Mathematics, 18 (2019), 197-202.   Google Scholar [13] C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.  Google Scholar [14] S. W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communications, Cryptography and Radar Applications, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar [15] P. Ke, J. Zhang and S. Zhang, On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2p^n$, Des. Codes Cryptogr., 67 (2013), 325-339.  doi: 10.1007/s10623-012-9610-9.  Google Scholar [16] Y. J. Kim, S. Y. Jin and H. Y. Song, Linear complexity and autocorrelation of prime cube sequences, AAECC, LNCS, 4851 (2007), 188-197.  doi: 10.1007/978-3-540-77224-8_23.  Google Scholar [17] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [18] M. B. Nathnson, Elementary Methods in Number Theory, Springer, Berlin, GTM 195, 2003.  Google Scholar [19] A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.   Google Scholar [20] Z. Xiao, X. Zeng, C. Li and T. Helleseth, New generalized cyclotomic binary sequences of period $p^2$, Des. Codes Cryptogr., 86 (2018), 1483-1497.  doi: 10.1007/s10623-017-0408-7.  Google Scholar [21] T. Yan, B. Huang and G. Xiao, Cryptographic properties of some binary generalized cyclotomic sequences with length $p^2$, Inf. Sci., 178 (2008), 1078-1086.  doi: 10.1016/j.ins.2007.02.040.  Google Scholar [22] T. Yan, S. Li and G. Xiao, On the linear complexity of generalized cyclotomic sequences with period $p^m$, Appl. Math. Lett., 21 (2008), 187-193.  doi: 10.1016/j.aml.2007.03.011.  Google Scholar [23] T. Yan and X. Li, Some notes on the generalized cyclotomic binary sequences of length $2p^m$ and $p^m$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 96 (2013), 2049-2051.  doi: 10.1007/s00200-012-0177-5.  Google Scholar [24] T. Yan, R. Sun and G. Xiao, Autocorrelation and linear complexity of the new generalized cyclotomic sequences, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 90 (2007), 857-864.   Google Scholar [25] X. Zeng, H. Cai, X. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.  Google Scholar [26] J. Zhang, C. Zhao and X. Ma, Linear complexity of generalized cyclotomic binary sequences of length $2p^m$, Applicable Algebra in Engineering Communication and Computing, 21 (2010), 93-108.  doi: 10.1007/s00200-009-0116-2.  Google Scholar

show all references

##### References:
 [1] E. Bai, X. Liu and G. Xiao, Linear complexity of new generalized cyclotomic sequences of order two of length $pq$, IEEE Trans. Inf. Theory, 51 (2005), 1849-1853.  doi: 10.1109/TIT.2005.846450.  Google Scholar [2] D.M. Burton, Elementary Number Theory, McGram-Hill, New York, 1998.  Google Scholar [3] T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004.  Google Scholar [4] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order $2$, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.  Google Scholar [5] C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inf. Theory, 44 (1998), 1699-1702.  doi: 10.1109/18.681354.  Google Scholar [6] C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.  doi: 10.1006/ffta.1998.0207.  Google Scholar [7] C. Ding and T. Helleseth, Generalized cyclotomy codes of length $p^{m_{1}}_{1}p^{m_{2}}_{2}\cdots p^{m_{t}}_{t}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.  doi: 10.1109/18.748996.  Google Scholar [8] X. Dong, Linear complexity of generalized cyclotomic binary sequences of length $4p^n$, Inf. Sci. Lett., 4 (2015), 67-70.   Google Scholar [9] V. Edemskiy, About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$, Des. Codes Cryptogr., 61 (2011), 251-260.  doi: 10.1007/s10623-010-9474-9.  Google Scholar [10] V. Edemskiy and O. Antonova, The evaluation of the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2^np^m$, International Journal of Mathematical Models and Methods in Applied Sciences, 9 (2015), 512-517.   Google Scholar [11] V. Edemskiy, C. Li, X. Zeng and T. Helleseth, The linear complexity of generalized cyclotomic binary sequences of period $p^n$, Des. Codes Cryptogr., 87 (2018), 1183-1197.  doi: 10.1007/s10623-018-0513-2.  Google Scholar [12] V. Edemskiy and C. Wu, On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo $2^np^m$, WSEAS Transactions on Mathematics, 18 (2019), 197-202.   Google Scholar [13] C. Fan and G. Ge, A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.  doi: 10.1109/TIT.2013.2290694.  Google Scholar [14] S. W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communications, Cryptography and Radar Applications, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar [15] P. Ke, J. Zhang and S. Zhang, On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2p^n$, Des. Codes Cryptogr., 67 (2013), 325-339.  doi: 10.1007/s10623-012-9610-9.  Google Scholar [16] Y. J. Kim, S. Y. Jin and H. Y. Song, Linear complexity and autocorrelation of prime cube sequences, AAECC, LNCS, 4851 (2007), 188-197.  doi: 10.1007/978-3-540-77224-8_23.  Google Scholar [17] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [18] M. B. Nathnson, Elementary Methods in Number Theory, Springer, Berlin, GTM 195, 2003.  Google Scholar [19] A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.   Google Scholar [20] Z. Xiao, X. Zeng, C. Li and T. Helleseth, New generalized cyclotomic binary sequences of period $p^2$, Des. Codes Cryptogr., 86 (2018), 1483-1497.  doi: 10.1007/s10623-017-0408-7.  Google Scholar [21] T. Yan, B. Huang and G. Xiao, Cryptographic properties of some binary generalized cyclotomic sequences with length $p^2$, Inf. Sci., 178 (2008), 1078-1086.  doi: 10.1016/j.ins.2007.02.040.  Google Scholar [22] T. Yan, S. Li and G. Xiao, On the linear complexity of generalized cyclotomic sequences with period $p^m$, Appl. Math. Lett., 21 (2008), 187-193.  doi: 10.1016/j.aml.2007.03.011.  Google Scholar [23] T. Yan and X. Li, Some notes on the generalized cyclotomic binary sequences of length $2p^m$ and $p^m$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 96 (2013), 2049-2051.  doi: 10.1007/s00200-012-0177-5.  Google Scholar [24] T. Yan, R. Sun and G. Xiao, Autocorrelation and linear complexity of the new generalized cyclotomic sequences, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 90 (2007), 857-864.   Google Scholar [25] X. Zeng, H. Cai, X. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.  doi: 10.1109/TIT.2013.2237754.  Google Scholar [26] J. Zhang, C. Zhao and X. Ma, Linear complexity of generalized cyclotomic binary sequences of length $2p^m$, Applicable Algebra in Engineering Communication and Computing, 21 (2010), 93-108.  doi: 10.1007/s00200-009-0116-2.  Google Scholar
The autocorrelation distribution of the binary sequence of period $108$
 $R_{S^{(d,e)}}(\tau)$ $\tau$ $-68$ $1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97,107$ $32$ $2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98,106$ $-4$ $3, 9, 33, 39, 69, 75, 99,105$ $-32$ $4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92,100,104$ $68$ $5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91,101,103$ $-80$ $6, 30, 42, 66, 78,102$ $80$ $12, 24, 48, 60, 84, 96$ $4$ $15, 21, 45, 51, 57, 63, 87, 93,99$ $-96$ $18, 90$ $0$ $27, 81$ $96$ $36, 72$ $-104$ $54$
 $R_{S^{(d,e)}}(\tau)$ $\tau$ $-68$ $1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97,107$ $32$ $2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98,106$ $-4$ $3, 9, 33, 39, 69, 75, 99,105$ $-32$ $4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92,100,104$ $68$ $5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91,101,103$ $-80$ $6, 30, 42, 66, 78,102$ $80$ $12, 24, 48, 60, 84, 96$ $4$ $15, 21, 45, 51, 57, 63, 87, 93,99$ $-96$ $18, 90$ $0$ $27, 81$ $96$ $36, 72$ $-104$ $54$
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