Multi-sequence linear shift-register synthesis: The varying length case

The problem of Linear shift-register synthesis for a single sequence is solved by the well known Berlekamp-Massey algorithm. The problem of multi-sequence shift-register synthesis is already addressed by Feng and Tzeng. The Feng-Tzeng algorithm can be considered as a generalization of the Berlekamp-Massey algorithm which takes a set of t different sequences of length N and yields a linear shift-register of length l capable of generating all t sequences. However, for the case of multiple sequences of varying length, the Feng-Tzeng algorithm generally does not give the correct solution. We demonstrate this by means of an example and explain, why the Feng-Tzeng algorithm does not work properly in the unequal length case. We propose a modification of the Fundamental Iterative Algorithm (FIA) from Feng and Tzeng, which overcomes the problem with varying length sequences. Based on this algorithm we derive an efficient Berlekamp-Massey like algorithm for solving the multisequence shift-register synthesis problem for sequences of varying length.