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 $2^{l-1}\le k<2^{l}$.