Feasible regions of symmetric capacity and Gallager´s E0 function for ternary-input discrete memoryless channels

In the refinement of a channel coding theorem, error exponents characterize the exponential convergence rates of decoding error probabilities. Error exponents are sometime called as reliability functions. In this study, we consider analyzing the reliability functions based on Gallager´s E₀ function. The region of the E₀ function of binary-input memoryless and symmetric channels for a fixed capacity was clarified by Guillén i Fàbregas et al. in their 2013 study. More precisely, binary erasure and binary symmetric channels have maximal and minimal E₀ functions, respectively, among the binary-input memoryless and symmetric channels for a fixed capacity. In this study, we extend their results from binary- to ternary-input channels that are not necessarily symmetric. First, we identify the extreme channels among the ternary-input strongly symmetric channels. Next, we identify the extreme channels among the ternary-input memoryless and symmetric channels. In addition, using channel symmetrization, we investigate whether the feasible regions of symmetric capacity and the E₀ functions for discrete memoryless channels (DMCs) are identical to those for symmetric channels if the channel inputs follow a uniform distribution. We describe the feasible regions for ternary-input DMCs under a uniform input distribution. In particular, we reveal the channels with maximal E₀ function among ternary-input DMCs for a fixed symmetric capacity and uniform input distribution.