The vector key equation and multisequence shift register synthesis

Abstract

We introduce the vector key equation for cyclic codes with more than one set of consecutive roots. Using the lattice basis reduction multisequence shift register synthesis algorithm [7,8], we find the minimal solution to the vector equation key and also give an important parameter of the multiple sequences, that is, characteristic sequence. Using it, we give a simpler sufficient and necessary condition for the uniqueness of the minimal polynomial of multiple sequences than that in [7]. The new approach enables us not only to decode cyclic codes up to HT and Roos bounds and but also to find out certain error patterns which can be decoded beyond the two bounds. © Springer-Verlag Berlin Heidelberg 2006.