Achieving the optimal guaranteed error correction rate in a simple concatenated coding scheme with single-trial decoding

For a concatenated coding scheme using a single inner code, a single outer code, and an optimal single-trial decoding strategy, we study the guaranteed error correction rate, i.e., the number of correctable errors per transmitted symbol. In particular, it is investigated whether the optimal guaranteed error correction rate can be achieved by finite length codes or only be approached by an infinite sequence of codes of increasing lengths. It is shown that optimizing the guaranteed error correction rate for a given inner or outer code does not necessarily mean pushing the code rate to zero. Furthermore, to optimize the guaranteed error correction rate over all inner and outer codes of fixed dimensions and alphabets, the code rate of one (but not both) of these two codes should be pushed to zero.