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

Linear Universal Decoding for Compound Channels: a Local to Global Geometric Approach