The structure and enumeration of periodic binary sequences with high nonlinear complexity

TL;DR

This paper characterizes the structure of high nonlinear complexity periodic binary sequences and provides an exact enumeration formula for sequences with nonlinear complexity at least three-quarters of their period.

Contribution

It offers a detailed structural analysis and an exact counting method for n-periodic binary sequences with high nonlinear complexity.

Findings

Characterized the structure of sequences with nonlinear complexity ≥ 3n/4

Derived an exact enumeration formula for these sequences

Enhanced understanding of sequence complexity and enumeration methods

Abstract

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary sequences with nonlinear complexity larger than or equal to 3n/4 is characterized. Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.