Enumeration of minimal convolutional encoders (Corresp.)

A general ensemble of all rate convolutional encoders with overall constraint length (OCL) is considered, and the notion of a minimum overall constraint length (MOCL) encoder is introduced. The noncatastrophic subclass of this class of zero delay MOCL encoders is the class of minimal encoders. Using this fact, upper and lower bounds on the number of rate minimal encoders are obtained. Results for finite values of and asymptotic results for large are obtained. In all cases (except in the limit as for rate the lower bound is a nonzero fraction of the upper bound, which in turn is a nonzero fraction of the total number of encoders. This implies that for an arbitrary fixed rate and for any OCL, a nonzero fraction of encoders is minimal, and hence noncatastrophic. In general, as the rate decreases the fraction of all zero delay encoders that are minimal increases. The upper and lower bounds tend to merge in the low rate case providing a good estimate of the number of minimal encoders. The results in the high rate case are not as satisfactory since the upper and lower bounds are widely separated. The results also indicate that for a given rate the probability of selecting a minimal encoder increases with increasing . Thus there may be some advantages to considering encoders with an increased number of input and output sequences.