Extremality Between Symmetric Capacity and Gallager’s Reliability Function $E_{0}$ for Ternary-Input Discrete Memoryless Channels

This paper examines the exact ranges between the symmetric capacity and Gallager's reliability function E
₀
for ternary-input discrete memoryless channels (T-DMCs) under a uniform input distribution. We first derive the two extremal ternary-input strongly symmetric channels taking the maximum and minimum values of the E
₀
function among all ternary-input strongly symmetric channels with a fixed capacity. Extending the results of ternary-input strongly symmetric channels, we second derive the exact ranges between capacity and the E
₀
function for ternary-input Gallager-symmetric channels. We third show that the exact ranges between the symmetric capacity and the E
₀
function of T-DMCs coincide with the ranges of ternaryinput Gallager-symmetric channels. In particular, we identify the extremal channels taking the maximum and minimum of E
₀
among all T-DMCs with a fixed symmetric capacity. As applications of the results, we describe some bounds of error exponents for T-DMCs with a fixed symmetric capacity.