Correlation distribution of a sequence family generalizing some sequences of Trachtenberg

Abstract

<p style='text-indent:20px;'>In this paper, we give a classification of a sequence family, over arbitrary characteristic, adding linear trace terms to the function <inline-formula><tex-math id="M1">\begin{document}$ g(x) = \mathrm{Tr}(x^d) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ d = p^{2k}-p^k+1 $\end{document}</tex-math></inline-formula>, first introduced by Trachtenberg. The family has <inline-formula><tex-math id="M3">\begin{document}$ p^n+1 $\end{document}</tex-math></inline-formula> cyclically distinct sequences with period <inline-formula><tex-math id="M4">\begin{document}$ p^n-1 $\end{document}</tex-math></inline-formula>. We compute the exact correlation distribution of the function <inline-formula><tex-math id="M5">\begin{document}$ g(x) $\end{document}</tex-math></inline-formula> with linear <inline-formula><tex-math id="M6">\begin{document}$ m $\end{document}</tex-math></inline-formula>-sequences and amongst themselves. The cross-correlation values are obtained as <inline-formula><tex-math id="M7">\begin{document}$ C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\} $\end{document}</tex-math></inline-formula>.</p>