  Achievable and converse bounds for general channels and mismatched decoding
are derived. The direct (achievable) bound is derived using random coding and
the analysis is tight up to factor 2. The converse is given in term of the
achievable bound and the factor between them is given. This gives performance
of the best rate-R code with possible mismatched decoding metric over a general
channel, up to the factor that is identified. In the matched case we show that
the converse equals the minimax meta-converse of Polyanskiy et al.
