Structure Analysis on the $k$-error Linear Complexity for $2^n$-periodic Binary Sequences

TL;DR

This paper characterizes the critical points of k-error linear complexity for 2^n-periodic binary sequences using cube theory, and provides complete counting functions for these complexities, advancing understanding of sequence structure.

Contribution

It introduces a novel cube theory-based decomposition and characterizes the second and third descent points of k-error linear complexity for the first time.

Findings

Complete characterization of the i-th descent points for i=2,3.

Distribution of the second descent point for k-error linear complexity.

Counting functions for the second descent point when k=3,4.

Abstract

In this paper, in order to characterize the critical error linear complexity spectrum (CELCS) for $2^n$-periodic binary sequences, we first propose a decomposition based on the cube theory. Based on the proposed $k$-error cube decomposition, and the famous inclusion-exclusion principle, we obtain the complete characterization of $i$th descent point (critical point) of the k-error linear complexity for $i=2,3$. Second, by using the sieve method and Games-Chan algorithm, we characterize the second descent point (critical point) distribution of the $k$-error linear complexity for $2^n$-periodic binary sequences. As a consequence, we obtain the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences as the second descent point for $k=3,4$. This is the first time for the second and the third descent points to be completely characterized. In fact, the proposed constructive approach has the potential to be used for constructing $2^n$-periodic binary sequences with the given linear complexity and $k$-error linear complexity (or CELCS), which is a challenging problem to be deserved for further investigation in future.