Periodic sequences with stable $k$-error linear complexity

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, 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 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 − (2 − 1) over all 2-periodic binary sequences, where 2 ≤ k < 2.