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Quasi-symmetric designs on $ 56 $ points
Correlation distribution of a sequence family generalizing some sequences of Trachtenberg
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 |
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\} $.
References:
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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. |
[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. |
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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. |
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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. |
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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. |
[6] |
R. Gold,
Optimal binary sequences for spread spectrum multiplexing, IEEE Trans. Inform. Theory, 13 (1967), 154-156.
doi: 10.1109/TIT.1967.1054048. |
[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. |
[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. |
[9] |
T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Vol. II, North-Holland, Amsterdam, 1998. |
[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. |
[11] |
R. Lidl and H. Niederreiter, Sequences with low correlation, in Finite Fields, Vol. 20, Cambridge University Press, Cambridge, 1997. |
[12] |
Y. Niho, Multi-Valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D thesis, University of Southern California, 1972. |
[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. |
[14] |
H. M. Trachtenberg, On the Cross-Correlation Function of Maximal Linear Recurring Sequences, Ph.D thesis, University of Southern California, 1970. |
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. |
[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. |
[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. |
[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. |
[5] |
J. F. Dillon,
Multiplicative difference sets via additive characters, Des. Codes Cryptogr., 17 (1999), 225-235.
doi: 10.1023/A:1026435428030. |
[6] |
R. Gold,
Optimal binary sequences for spread spectrum multiplexing, IEEE Trans. Inform. Theory, 13 (1967), 154-156.
doi: 10.1109/TIT.1967.1054048. |
[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. |
[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. |
[9] |
T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, Vol. II, North-Holland, Amsterdam, 1998. |
[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. |
[11] |
R. Lidl and H. Niederreiter, Sequences with low correlation, in Finite Fields, Vol. 20, Cambridge University Press, Cambridge, 1997. |
[12] |
Y. Niho, Multi-Valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, Ph.D thesis, University of Southern California, 1972. |
[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. |
[14] |
H. M. Trachtenberg, On the Cross-Correlation Function of Maximal Linear Recurring Sequences, Ph.D thesis, University of Southern California, 1970. |
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