Further Results on the Distinctness of Decimations of l-sequences

Let $\underline{a}$ be an \textit{l}-sequence generated by a feedback-with-carry shift register with connection integer $q=p^{e}$, where $p$ is an odd prime and $e\geq 1$. Goresky and Klapper conjectured that when $p^{e}\notin \{5,9,11,13\}$, all decimations of $\underline{a}$ are cyclically distinct. When $e=1$ and $p>13$, they showed that the set of distinct decimations is large and, in some cases, all deciamtions are distinct. In this article, we further show that when $e\geq 2$ and$p^{e}\neq 9$, all decimations of $\underline{a}$ are also cyclically distinct.Comment: submitted to IEEE-I