Refined Convergents to the Associated Continued Fractions for Binary Sequences

The relation between continued fractions and Berlekamp's algorithm was studied by some reseachers. The latter is an iterative procedure proposed for decoding BCH codes. However, there remains an unanswered question whether each of the iterative steps in the algorithm can be interpreted in terms of continued fractions. In this paper, we first introduce the so-called refined convergents to the continued fraction expansion of a binary sequence S, and then give a thorough answer to the question in the context of Massey's linear feedback shift register synthesis algorithm which is equivalent to that of Berlekamp, and at last we prove that there exists a one- to-one correspondence between the n-th refined convergents and the length n segments