# 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
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