مرکز منطقه ای اطلاع رساني علوم و فناوري - A fundamental inequality between the probabilities of binary subgroups and cosets

Abstract :

The probability of a set of binary $n$ -tuples is defined to be the sum of the probabilities of the individual $n$ -tuples when each digit is chosen independently with the same probability $p$ of being a "one." It is shown that, under such a definition, the ratio between the probability of a subgroup of order $2^{k}$ and any of its proper cosets is always greater than or equal to a function $F_{k}(p)$ , where $F_{k}(p) \\geq 1$ for $p \\leq frac{1}{2}$ with equality when and only when $p = frac{1}{2}$ . It is further shown that $F_{k}(p)$ is the greatest lower bound on this ratio, since a subgroup and proper coset of order $2^{k}$ can always be found such that the ratio between their probabilities is exactly $F_{k}(p)$ . 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.