A fundamental inequality between the probabilities of binary subgroups and cosets

The probability of a set of binary -tuples is defined to be the sum of the probabilities of the individual -tuples when each digit is chosen independently with the same probability of being a "one." It is shown that, under such a definition, the ratio between the probability of a subgroup of order and any of its proper cosets is always greater than or equal to a function , where for with equality when and only when . It is further shown that is the greatest lower bound on this ratio, since a subgroup and proper coset of order can always be found such that the ratio between their probabilities is exactly . 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.