On the computation of the linear complexity and the k-error linear complexity of binary sequences with period a power of two

The linear Games-Chan algorithm for computing the linear complexity c(s) of a binary sequence s of period l = 2(n) 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 and of the error linear complexity spectrum for finite binary sequences viewed as initial segments of infinite sequences with period a power of two. Lauder and Paterson apply their algorithm to decoding binary repeated-root cyclic codes of length l = 2(n) in O(l(log(2) l)(2)) time. We improve on their result, developing a decoding algorithm with O(l) bit complexity.