  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 $2^n$, one could convert the computation of
$k$-error linear complexity into finding error sequences with minimal Hamming
weight. Based on Games-Chan algorithm, the $k$-error linear complexity
distribution of $2^n$-periodic binary sequences is investigated in this paper.
First, for $k=2,3$, 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 characterized. Second, for $k=3,4$, the complete counting
functions on the $k$-error linear complexity of $2^n$-periodic binary sequences
with linear complexity $2^n$ are presented. Third, as a consequence of these
results, the counting functions for the number of $2^n$-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.
