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

• Authors: R. Ahlswede;Ning Cai;Zhen ZHang
• Venue: IEEE Transactions on Information Theory
• Year: 2006

For the discrete memoryless channel (χ, y, W) we give characterizations of the zero-error erasure capacity Cer and the zero-error average list size capacity Cal 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 Wε, where for y +(x)={y:W(y|x)>0} Wε(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 Cer(Wε) and Cal (Wε) 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 Coid for identification codes with zero misrejection probability and misacceptance probability tending to zero. Our Theorem 3 says that Coid equals the zero-error erasure capacity for transmission Cer