Oblivious Communication Channels and Their Capacity

Let be C={x₁,...,x_n} sub {0,1}ⁿ be an [n,N] binary error correcting code (not necessarily linear). Let e{0,1}ⁿ be an error vector. A codeword xepsiC is said to be disturbed by the error e if the closest codeword to xopluse is no longer x. Let A_e be the subset of codewords in C that are disturbed by e. In this work, we study the size of A_e in random codes C (i.e., codes in which each codeword is chosen uniformly and independently at random from {0,1}ⁿ ). Using recent results of Vu [Random Structures and Algorithms, vol. 20, no. 3, pp. 262-316, 2002] on the concentration of non-Lipschitz functions, we show that |A_e| is strongly concentrated for a wide range of values of N and ||e||. We apply this result in the study of communication channels we refer to as oblivious. Roughly speaking, a channel W(y|x) is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x. A family of channels Psi is said to be oblivious if every member W of the family is oblivious. In this work, we define oblivious and partially oblivious families of (not necessarily memoryless) channels and analyze their capacity. When considering the capacity of a family of channels Psi, one must address the design of error correcting codes which allow communication under the uncertainty of which channel WepsiPsi is actually used. The oblivious channels we define have connections to arbitrarily varying channels with state constraints.