Tables of graphs of binary and ternary sequences differentiation

TL;DR

This paper provides graphical representations of the dynamic system of difference operations on cyclic sequences over finite fields, aiding in the analysis of Arnold hypotheses for binary and ternary sequences.

Contribution

It presents detailed graphs of the difference operation dynamics for sequences over finite fields, supporting the analysis and proof of Arnold hypotheses.

Findings

Graphs of the difference operation system for sequences over =2, n300

Graphs of the difference operation system for sequences over =3, n150

Supports proof and analysis of Arnold hypotheses

Abstract

Let $x$ be a cyclic sequence of $n$ elements of the finite field $\mathbb{F}_q$ (the first element immediately follows the $n$-th one). Let us define the operation $\Delta$ as the transition from $x$ to the sequence of differences of the neighbouring elements from $x$. The aim of this work is to give graphs of the dynamic system $\Delta$ for $q=2$, $n\le 300$ and $q=3$, $n\le 150$. These results enable us to define more precisely the Arnold hypotheses and to prove them.