Characterization of $2^n$-periodic binary sequences with fixed 3-error or 4-error linear complexity

The linear complexity and the k-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, the k-error linear complexity distribution of 2-periodic binary sequences is investigated based on Games-Chan algorithm. First, for k = 2, 3, the complete counting functions on the k-error linear complexity of 2-periodic binary sequences with linear complexity less than 2 are characterized. Second, for k = 3, 4, the complete counting functions on the k-error linear complexity of 2-periodic binary sequences with linear complexity 2 are presented. Third, for k = 4, 5, the complete counting functions on the k-error linear complexity of 2-periodic binary sequences with linear complexity less than 2 are derived. As a consequence of these results, the counting functions for the number of 2periodic binary sequences with the 3-error linear complexity are obtained, and the complete counting functions on the 4-error linear complexity of 2-periodic binary sequences are obvious.