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November  2021, 15(4): 647-662. doi: 10.3934/amc.2020087

Correlation distribution of a sequence family generalizing some sequences of Trachtenberg

Ferruh Özbudak 1, and  Eda Tekin 2,3,,

1. 

Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey
2. 

Department of Business Administration, Karabük University, 78050, Karabük, Turkey
3. 

Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

* Corresponding author: Eda Tekin

Received  November 2019 Published  November 2021 Early access  June 2020

In this paper, we give a classification of a sequence family, over arbitrary characteristic, adding linear trace terms to the function $ g(x) = \mathrm{Tr}(x^d) $, where $ d = p^{2k}-p^k+1 $, first introduced by Trachtenberg. The family has $ p^n+1 $ cyclically distinct sequences with period $ p^n-1 $. We compute the exact correlation distribution of the function $ g(x) $ with linear $ m $-sequences and amongst themselves. The cross-correlation values are obtained as $ C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\} $.

Keywords: Sequences, cross-correlation, plateaued functions.
Mathematics Subject Classification: Primary: 94A55, 11T71; Secondary: 14G50.
Citation: Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2021, 15 (4) : 647-662. doi: 10.3934/amc.2020087
References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235–-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[2]

S. Boztaş and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inform. Theory, 40 (1994), 532-537.  doi: 10.1109/18.312181.  Google Scholar

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S. Boztaş, F. Özbudak and E. Tekin, Explicit full correlation distribution of sequence families using plateaued functions, IEEE Trans. Inform. Theory, 64 (2018), 2858-2875.  doi: 10.1109/TIT.2017.2789208.  Google Scholar

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E. Çakçak and F. Özbudak, Some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places, J. Pure Appl. Algebra, 210 (2007), 113-135.  doi: 10.1016/j.jpaa.2006.08.007.  Google Scholar

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J. F. Dillon, Multiplicative difference sets via additive characters, Des. Codes Cryptogr., 17 (1999), 225-235.  doi: 10.1023/A:1026435428030.  Google Scholar

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R. Gold, Optimal binary sequences for spread spectrum multiplexing, IEEE Trans. Inform. Theory, 13 (1967), 154-156.  doi: 10.1109/TIT.1967.1054048.  Google Scholar

[7]

R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE Trans. Inform. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106.  Google Scholar

[8]

T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar

[9]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Vol. II, North-Holland, Amsterdam, 1998.  Google Scholar

[10]

T. Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Information and Control, 18 (1971), 369-394.  doi: 10.1016/S0019-9958(71)90473-6.  Google Scholar

[11]

R. Lidl and H. Niederreiter, Sequences with low correlation, in Finite Fields, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar

[12]

Y. Niho, Multi-Valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D thesis, University of Southern California, 1972. Google Scholar

[13]

X. Tang, T. Helleseth, L. Hu and W. Jiang, A new family of Gold-like sequences, Sequences, Subsequences, and Consequences, Lecture Notes in Comput. Sci., Vol. 4893, Springer, Berlin, 2007, 62–69. doi: 10.1007/978-3-540-77404-4_6.  Google Scholar

[14]

H. M. Trachtenberg, On the Cross-Correlation Function of Maximal Linear Recurring Sequences, Ph.D thesis, University of Southern California, 1970. Google Scholar

show all references

References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235–-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[2]

S. Boztaş and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inform. Theory, 40 (1994), 532-537.  doi: 10.1109/18.312181.  Google Scholar

[3]

S. Boztaş, F. Özbudak and E. Tekin, Explicit full correlation distribution of sequence families using plateaued functions, IEEE Trans. Inform. Theory, 64 (2018), 2858-2875.  doi: 10.1109/TIT.2017.2789208.  Google Scholar

[4]

E. Çakçak and F. Özbudak, Some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places, J. Pure Appl. Algebra, 210 (2007), 113-135.  doi: 10.1016/j.jpaa.2006.08.007.  Google Scholar

[5]

J. F. Dillon, Multiplicative difference sets via additive characters, Des. Codes Cryptogr., 17 (1999), 225-235.  doi: 10.1023/A:1026435428030.  Google Scholar

[6]

R. Gold, Optimal binary sequences for spread spectrum multiplexing, IEEE Trans. Inform. Theory, 13 (1967), 154-156.  doi: 10.1109/TIT.1967.1054048.  Google Scholar

[7]

R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE Trans. Inform. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106.  Google Scholar

[8]

T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar

[9]

T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Vol. II, North-Holland, Amsterdam, 1998.  Google Scholar

[10]

T. Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Information and Control, 18 (1971), 369-394.  doi: 10.1016/S0019-9958(71)90473-6.  Google Scholar

[11]

R. Lidl and H. Niederreiter, Sequences with low correlation, in Finite Fields, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar

[12]

Y. Niho, Multi-Valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D thesis, University of Southern California, 1972. Google Scholar

[13]

X. Tang, T. Helleseth, L. Hu and W. Jiang, A new family of Gold-like sequences, Sequences, Subsequences, and Consequences, Lecture Notes in Comput. Sci., Vol. 4893, Springer, Berlin, 2007, 62–69. doi: 10.1007/978-3-540-77404-4_6.  Google Scholar

[14]

H. M. Trachtenberg, On the Cross-Correlation Function of Maximal Linear Recurring Sequences, Ph.D thesis, University of Southern California, 1970. Google Scholar

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