On the Computation of the Linear Complexity and the k-Error Linear Complexity of Binary Sequences with Period a Power of Two

Abstract

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 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 = 2ⁿ in ${\mathcal O}(\ell({\rm log}_{2}\ell)^2)$ time. We improve on their result, developing a decoding algorithm with ${\mathcal O}(\ell)$ bit complexity.