The 4-error linear complexity distribution for $2^n$-periodic binary sequences

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