  Over discrete memoryless channels (DMC), linear decoders (maximizing additive
metrics) afford several nice properties. In particular, if suitable encoders
are employed, the use of decoding algorithm with manageable complexities is
permitted. Maximum likelihood is an example of linear decoder. For a compound
DMC, decoders that perform well without the channel's knowledge are required in
order to achieve capacity. Several such decoders have been studied in the
literature. However, there is no such known decoder which is linear. Hence, the
problem of finding linear decoders achieving capacity for compound DMC is
addressed, and it is shown that under minor concessions, such decoders exist
and can be constructed. This paper also develops a "local geometric analysis",
which allows in particular, to solve the above problem. By considering very
noisy channels, the original problem is reduced, in the limit, to an inner
product space problem, for which insightful solutions can be found. The local
setting can then provide counterexamples to disproof claims, but also, it is
shown how in this problem, results proven locally can be "lifted" to results
proven globally.
