The Asymptotic Optimum Properties of Group and Systematic Codes for Some Channels

Memoryless stationary channels are considered having symbols $E_1,\dots,E_M$, which are letters of the alphabet of the channel, and $M=s^k$, $s$ being a prime number and $k$ an integer. It is supposed that the symbols $E_1,\dots,E_M$ are the elements of a commutative group and, moreover, the transition probability from $E_i$ to $E_j$ coincides with the transition probability from $E_i+E_k$ to $E_j+E_k$ for all indeces $I$, $j$, $k$. 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 present paper.