Erasure, list, and detection zero-error capacities for low noise and a relation to identification

For the discrete memoryless channel (χ, y, W) we give characterizations of the zero-error erasure capacity C_er and the zero-error average list size capacity C_al in terms of limits of suitable information (respectively, divergence) quantities (Theorem 1). However, they do not “single-letterize.” Next we assume that χ⊂y and W(x|x)>0 for all x∈χ, and we associate with W the low-noise channel W_ε, where for y ⁺(x)={y:W(y|x)>0} W_ε(y|x)={1, if y=x and |y⁺(x)|=1 1-ε, if y=x and |y⁺(x)|>1 e/|y ⁺(x)|-1, if y≠x. Our Theorem-2 says that as ε tends to zero the capacities C_er(W_ε) and C_al (W_ε) relate to the zero-error detection capacity C _de(W). Our third result is a seemingly basic contribution to the theory of identification via channels. We introduce the (second-order) identification capacity C_oid for identification codes with zero misrejection probability and misacceptance probability tending to zero. Our Theorem 3 says that C_oid equals the zero-error erasure capacity for transmission C_er