Characterization of the Third Descent Points for the k-error Linear Complexity of $$2^n$$-periodic Binary Sequences

Abstract

In this paper, a structural approach for determining CELCS (critical error linear complexity spectrum) for the k-error linear complexity distribution of \(2^n\)-periodic binary sequences is developed via the sieve method and Games-Chan algorithm. Accordingly, the third descent point (critical point) distribution of the k-error linear complexity for \(2^n\)-periodic binary sequences is characterized. As a consequence, we derive the complete counting functions on the 5-error linear complexity of \(2^n\)-periodic binary sequences when it is the third descent point. With the structural approach proposed here, one can further characterize other third and fourth descent points of the k-error linear complexity for \(2^n\)-periodic binary sequences.