# American Institute of Mathematical Sciences

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

## 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

* 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 $C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\}$.

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:
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##### 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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