Comlexity of prime-dimensional sequences over a finite field

TL;DR

This paper investigates the complexity of sequences over finite fields of prime dimension, showing that most sequences exhibit high complexity as the dimension grows, with specific results on multiplicative functions and logarithmic sequences.

Contribution

It characterizes the complexity of sequences over finite fields of prime dimension, proving that high complexity is typical and identifying special cases like multiplicative functions and logarithmic sequences.

Findings

Almost all sequences have high complexity as dimension n increases.

Sequences from multiplicative functions are complex for all n not equal to the field's characteristic.

Logarithmic sequences are nearly most complicated for certain prime values of n.

Abstract

V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a finite field, whose dimension $n$ is a prime number. We prove that with $n\to \infty$ this property is inherent in almost all sequences, while the values of multiplicative functions possess this property with any $n$ different from the characteristic of the field. We also describe the prime values of the parameter $n$ which make the logarithmic function almost most complicated. All these sequences reveal a stronger complexity; its algebraic sense is quite clear.