Minimum distance estimates of the performance of sequential decoders

TLDR: Simple tests are derived that allow easy determination of the performance on the BSC of a given binary convolutional code decoded with a modified version of the Fano algorithm.

Abstract: In the past, criteria for predicting the performance of individual codes with sequential decoding have been intuitive. In this paper, simple tests are derived that allow easy determination of the performance on the BSC (binary symmetric channel) of a given binary convolutional code decoded with a modified version of the Fano algorithm. A "distance-guaranteed computational cutoff rate," R_{dgcomp} , is defined in terms of the BSC crossover probability and the "uniform minimum distance" of the code. The latter is a measure of the minimum distance between codewords of all lengths up to and including the constraint length of the code. A bound is derived on the average number of decoding computations and is shown to be small and insensitive to constraint length if the code rate, R , satisfies the test R . Also, the probability of a decoding error is overbounded and the bound decreases exponentially with constraint length with exponent (R_{dgcomp} - R) . Consequently, the probability of error is small if (R_{dgcom} - R) is large. The existence of binary convolutional codes with a uniform minimum distance which meets the Gilbert bound is demonstrated. This result is combined with the condition R to show the existence of codes of rate less than a rate R_{D} for which the average number of decoding computations is small. The rate R_{D} is approximately one half of the true computational cutoff rate R_{comp} on the BSC with crossover probability of 10^{-4} .