SALAGEAN, A.M., 2005. On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two. IEEE transactions on information theory 51 (3) ,pp. 1145-1150
The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period ℓ = 2n requires the knowledge of the full sequence, while the quadratic Berlekamp-Massey algorithm only requires knowledge of 2c(s) terms. We show that we can modify the Games-Chan algorithm so that it computes the complexity in linear time knowing only 2c(s) terms. The algorithms of Stamp-Martin and Lauder-Paterson can also be modified, without loss of efficiency, to compute analogues of the k-error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two.
We also develop an algorithm which, given a constant c and an infinite
binary sequence s with period ℓ = 2n, computes the minimum number k of errors (and the associated error sequence) needed over a period
of s for bringing the linear complexity of s below c. The algorithm has a time and space bit complexity of O(ℓ). We apply our algorithm to
decoding and encoding binary repeated-root cyclic codes of length ℓ in linear, O(ℓ), time and space. A previous decoding algorithm proposed
by Lauder and Paterson has O(ℓ(logℓ)2) complexity.