  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.
