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

Abstract

For the discrete memoryless channel (/spl chi/, y, W) we give characterizations of the zero-error erasure capacity C/sub er/ and the zero-error average list size capacity C/sub al/ in terms of limits of suitable information (respectively, divergence) quantities (Theorem 1). However, they do not "single-letterize." Next we assume that /spl chi//spl sub/y and W(x|x)>0 for all x/spl isin//spl chi/, and we associate with W the low-noise channel W/sub /spl epsiv//, where for y/sup +/(x)={y:W(y|x)>0} W/sub /spl epsiv//(y|x)={1, if y=x and |y/sup +/(x)|=1 1-/spl epsiv/, if y=x and |y/sup +/(x)|>1 e/|y/sup +/(x)|-1, if y/spl ne/x. Our Theorem-2 says that as /spl epsi/ tends to zero the capacities C/sub er/(W/sub /spl epsi//) and C/sub al/(W/sub /spl epsi//) relate to the zero-error detection capacity C/sub 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/sub oid/ for identification codes with zero misrejection probability and misacceptance probability tending to zero. Our Theorem 3 says that C/sub oid/ equals the zero-error erasure capacity for transmission C/sub er/.