Correlation, Linear Complexity, Maximum order Complexity on Families of binary Sequences

TL;DR

This paper investigates the relationship between correlation measures and two pseudorandom measures, linear and maximum order complexity, in binary sequences, providing improved bounds and insights into their dependence.

Contribution

It introduces simplified and improved lower bounds for linear and maximum order complexities based on correlation measures, enhancing understanding of sequence pseudorandomness.

Findings

Improved lower bounds for linear complexity

Enhanced bounds for maximum order complexity

Clarified relation between correlation and complexity measures

Abstract

Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order $k$ and another two pseudorandom measures: the $N$th linear complexity and the $N$th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.