Majority decoding of large repetition codes for the r-ary symmetric channel

Abstract

A r-ary symmetric channel has as transition probability matrix the r×r matrix qxy=p if x≠y and qxy=1−(r−1)p=q if x=y. Given a set Y of r symbols, the code here consists of r codewords, each one of them is made up of n identical symbols. Whenever q is larger than p, maximum likelihood decoding amounts to find out in the received vector which symbol is repeated most. Thus MLD here reduces to majority decoding.

A generating function for the error probability as well as the probability of decoding failure for the system is obtained. Also recurrence relations are given for computing those probabilities.

We more generally consider a DMC (Discrete Memoryless Channel) which we call transitive. A r-ary transitive DMC is a DMC such that there exists a transitive permutation group G on the set Y of symbols such that

$$\forall \left( {x, y} \right) \in YxY,\forall \sigma \in G:q_{xy} = q_{\sigma x,\sigma y} .$$

The results corresponding to those announced for the r-ary symmetric channel are obtained for the majority decoding repetition codes over r-ary transitive DMC.