Comlexity of prime-dimensional sequences over a finite field

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.