Description of Binary Sequences Based on the Interval Linear Complexity Profile

Summary

We consider the problem of partial approximation of binary sequences by the outputs of linear feedback shift registers. A generalization of the linear complexity profiles of binary sequences leads to a sequence that is regarded as the profile of interval linear complexity. This concept is useful if we want to identify all fragments of the given binary sequence having length at least 2m which can be described by m + n bits, where n < m. We show that a fixed j-th component belongs to such a fragment with probability ~ 1/3 in the ensemble of fair coin-tossing sequences. We also apply the interval linear complexity approach to identify corrupted positions of the sequence generated by a linear feedback shift register with known feedback polynomial, where “corruption” of the sequence means either deletion, insertion or inversion of its components.