  The linear complexity of a sequence has been used as an important measure of
keystream strength, hence designing a sequence which possesses high linear
complexity and $k$-error linear complexity is a hot topic in cryptography and
communication. Niederreiter first noticed many periodic sequences with high
$k$-error linear complexity over GF(q). In this paper, the concept of stable
$k$-error linear complexity is presented to study sequences with high $k$-error
linear complexity. By studying linear complexity of binary sequences with
period $2^n$, the method using cube theory to construct sequences with maximum
stable $k$-error linear complexity is presented. It is proved that a binary
sequence with period $2^n$ can be decomposed into some disjoint cubes. The cube
theory is a new tool to study $k$-error linear complexity. Finally, it is
proved that the maximum $k$-error linear complexity is $2^n-(2^l-1)$ over all
$2^n$-periodic binary sequences, where $2^{l-1}\le k<2^{l}$.
