A CHARACTERIZATION OF CONVERGING DUCCI SEQUENCES OVER Z(2)

It is well-known that any Ducci sequence generated by a vector of length a power of 2 will eventually reach the null vector. As an easy consequence, all vectors obtained by concatenating several copies of a vector of length 2 eventually reach the null vector. We prove a converse to this statement for Ducci sequences over the field Z(2). Namely that over Z(2), the only vectors converging to the null vector are the vectors obtained by concatenation of several copies of a vector of length a power of 2.