Further Results on the Distinctness of Decimations of l-sequences

TL;DR

This paper proves that for certain l-sequences generated by feedback-with-carry shift registers, all their decimations are cyclically distinct, extending previous results to cases where the connection integer is a higher power of an odd prime.

Contribution

It extends the proof of the cyclic distinctness of decimations of l-sequences to cases where the connection integer is a higher power of an odd prime, specifically for e ≥ 2 and p^e ≠ 9.

Findings

All decimations are cyclically distinct for e ≥ 2 and p^e ≠ 9.

Confirms Goresky and Klapper's conjecture for broader cases.

Provides theoretical proof for the distinctness of decimations in these cases.

Abstract

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.