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

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 studying the linear complexity of binary sequences with period 2n, one could convert the computation of kerror linear complexity into finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, the k-error linear complexity distribution of 2n-periodic binary sequences is investigated in this paper. First, for k = 2, 3, the complete counting functions on the k-error linear complexity of 2n-periodic balanced binary sequences (with linear complexity less than 2n) are characterized. Second, for k = 3, 4, the complete counting functions on the k-error linear complexity of 2n-periodic binary sequences with linear complexity 2n are presented. Third, as a consequence of these results, the counting functions for the number of 2n-periodic binary sequences with the k-error linear complexity for k = 2 and 3 are obtained. Further more, an important result in a recent paper is proved to be not completely correct.