A fundamental inequality between the probabilities of binary subgroups and cosets

Abstract
The probability of a set of binaryn-tuples is defined to be the sum of the probabilities of the individualn-tuples when each digit is chosen independently with the same probabilitypof being a "one." It is shown that, under such a definition, the ratio between the probability of a subgroup of order2^{k}and any of its proper cosets is always greater than or equal to a functionF_{k}(p), whereF_{k}(p) \geq 1forp \leq \frac{1}{2}with equality when and only whenp = \frac{1}{2}. It is further shown thatF_{k}(p)is the greatest lower bound on this ratio, since a subgroup and proper coset of order2^{k}can always be found such that the ratio between their probabilities is exactlyF_{k}(p). It is then demonstrated that for a linear code on a binary symmetric channel the "tall-zero" syndrome is more probable than any other syndrome. This result is applied to the problem of error propagation in convolutional codes.

This publication has 2 references indexed in Scilit: