Asymptotic Optimality of Group and Systematic Codes for Some Channels

Memoryless stationary channels are considered having symbols E₁, , EM, which are letters of the alphabet of the channel, where M = sᵏ, s being a prime number and k an integer. It is supposed that the symbols E₁, , EM are the elements of a commutative group and, moreover, the transition probability from Eᵢ to Eⱼ coincides with the transition probability from Eᵢ + Eₖ to Eⱼ + Eₖ for all indices i, j, k. It is proved that in some sense the minimum probability of errors for all the codes is asymptotically equal to the minimum probability of error for all the group codes provided the transmission rate is large enough. Some other similar results are also proved in the present paper.