  By using the sieve method of combinatorics, we study $k$-error linear
complexity distribution of $2^n$-periodic binary sequences based on Games-Chan
algorithm. For $k=4,5$, the complete counting functions on the $k$-error linear
complexity of $2^n$-periodic balanced binary sequences (with linear complexity
less than $2^n$) are presented. As a consequence of the result, the complete
counting functions on the 4-error linear complexity of $2^n$-periodic binary
sequences (with linear complexity $2^n$ or less than $2^n$) are obvious.
Generally, the complete counting functions on the $k$-error linear complexity
of $2^n$-periodic binary sequences can be obtained with a similar approach.
