Correlation distribution of a sequence family generalizing some sequences of Trachtenberg
Ferruh Özbudak, Eda Tekin · Advances in Mathematics of Communications · 2020
In this paper, we give a classification of a sequence family, over arbitrary characteristic, adding linear trace terms to the function \begin{document}$ g(x) = \mathrm{Tr}(x^d) $\end{document} , where \begin{document}$ d = p^{2k}-p^k+1 $\end{document} , first introduced by Trachtenberg. The family has \begin{document}$ p^n+1 $\end{document} cyclically distinct sequences with period \begin{document}$ p^n-1 $\end{document} . We compute the exact correlation distribution of the function \begin{document}$ g(x) $\end{document} with linear \begin{document}$ m $\end{document} -sequences and amongst themselves. The cross-correlation values are obtained as \begin{document}$ C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\} $\end{document} .