Is there a way to construct a linear feedback shift register (LFSR) which outputs no more than k consecutive 1s or 0s? (It would have to be not a maximal-length LFSR, of course, if k < the degree of the characteristic polynomial)
edit: I am also looking for one where the period is at least 1/16 the maximum period of the LFSR.
(clarification: exclude the trivial case of all zeros. Also, I can initialize the LFSR to an arbitrary vector.)