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 lscr=2ⁿ requires the knowledge of the full sequence, while the quadratic Berlekamp-Massey algorithm requires knowledge of only 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 analogs 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 lscr=2ⁿ , computes the minimum number k of errors (and an 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(lscr). We apply our algorithm to decoding and encoding binary repeated-root cyclic codes of length lscr in linear, O(lscr), time and space. A previous decoding algorithm proposed by Lauder and Paterson has O(lscr(loglscr)²) complexity