Binary sequences with length n and nonlinear complexity not less than n/2

TL;DR

This paper develops an algorithm to generate and count binary sequences of length n with nonlinear complexity at least n/2, providing a complete distribution analysis of such sequences.

Contribution

It introduces a novel characterization and an algorithm for constructing all binary sequences with high nonlinear complexity, along with an exact counting formula.

Findings

Algorithm for generating sequences with nonlinear complexity ≥ n/2

Exact formula for counting these sequences

Complete distribution of nonlinear complexity for the sequences

Abstract

In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all binary sequences with length $n$ and nonlinear complexity $c_{n}\geq n/2$, where $n$ is an integer larger than $2$. Furthermore, a formula is established to calculate the exact number of these sequences. The distribution of nonlinear complexity for these sequences is thus completely determined.