A basic common characteristic of almost all channel coding problems treated in this book is that an asymptotically vanishing probability of error in transmission is tolerated. This permits us to exploit the global knowledge of the statistics of sources and channels in order to enhance transmission speed. We see again and again that in the case of a correct tuning of the parameters most codes perform in the same manner and thus, in particular, optimal codes, instead of being rare, abound. This ceases to be true if we are dealing with codes that are error-free.

The zero-error capacity of a DMC or compound DMC has been defined in Chapters 6 and 10 as the special case ε = 0 of ε-capacity. To keep this chapter self-contained, we give an independent (of course, equivalent) definition below.

A zero-error code of block length n for a DMC will be defined by a (codeword) set C ⊂ Xn, rather than by an encoder–decoder pair (f, φ), understanding that the message set coincides with the codeword set and the encoder is the identity mapping. This definition makes sense because if to a codeword set C there exists a decoder φ : Yn → C that yields probability of error equal to zero, this decoder is essentially unique.