This paper concerns itself with the question of list decoding for general
adversarial channels, e.g., bit-flip (XOR) channels, erasure
channels, AND (Z-) channels, OR channels, real adder
channels, noisy typewriter channels, etc. We precisely characterize when
exponential-sized (or positive rate) (L−1)-list decodable codes (where the
list size L is a universal constant) exist for such channels. Our criterion
asserts that:
"For any given general adversarial channel, it is possible to construct
positive rate (L−1)-list decodable codes if and only if the set of completely
positive tensors of order-L with admissible marginals is not entirely
contained in the order-L confusability set associated to the channel."
The sufficiency is shown via random code construction (combined with
expurgation or time-sharing). The necessity is shown by
1. extracting equicoupled subcodes (generalization of equidistant code) from
any large code sequence using hypergraph Ramsey's theorem, and
2. significantly extending the classic Plotkin bound in coding theory to list
decoding for general channels using duality between the completely positive
tensor cone and the copositive tensor cone. In the proof, we also obtain a new
fact regarding asymmetry of joint distributions, which be may of independent
interest.
Other results include
1. List decoding capacity with asymptotically large L for general
adversarial channels;
2. A tight list size bound for most constant composition codes
(generalization of constant weight codes);
3. Rederivation and demystification of Blinovsky's [Bli86] characterization
of the list decoding Plotkin points (threshold at which large codes are
impossible);
4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error
correction code setting