The k-error linear complexity of periodic sequences is an important security index of stream cipher systems. By using an interesting decomposing approach, we investigate the intrinsic structure for the set of 2n-periodic binary sequences with fixed complexity measures. For k ≤ 4, we construct the complete set of error vectors that give the k-error linear complexity. As auxiliary results we obtain the counting functions of the k-error linear complexity of 2n-periodic binary sequences for k ≤ 4, as well as the expectations of the k-error linear complexity of a random sequence for k ≤ 3. Moreover, we study the 2t-error linear complexity of the set of 2n-periodic binary sequences with some fixed linear complexity L, where t < n-1 and the Hamming weight of the binary representation of 2n-L is t. Also, we extend some results to pn-periodic sequences over Fp. Finally, we discuss some potential applications.