Abstract
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 (ℤ-) 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 essentially 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 by1. extracting approximately equicoupled subcodes (generalization of equidistant codes) from any sequence of "large" codes using hypergraph Ramsey's theorem, and2. 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 may be of independent interest.Other results include1 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 [9] characterization of the list decoding Plotkin points (threshold at which large codes are impossible) for bit-flip channels;4 Evaluation of general bounds ([43]) for unique decoding in the error correction code setting.
Highlights
While the main contribution of this work is to strictly generalize notions that have been primarily studied for “Hamming metric” channels, before we precisely define general channels, let us reprise what is known for Hamming metric channels . 51:52.1 Error correction and the Plotkin boundThe theory of error correction codes is about protecting data from errors
Generalizing [43], we show that codes with order-4 joint types Px1,x2,x3,x4 exist if and only if Px1,x2,x3,x4 is a completely positive tensor of order-4, i.e., Px1,x2,x3,x4 can be written as a convex combination of products of independent and identical distributions, k
To show that no large pL1q-list decodable code exists for general adversarial channels when Ppx1, ̈ ̈ ̈,xL is not completely positive, we provide upper and lower bounds on the average inner product between the empirical distribution of an L-tuple and a copositive witness of non-complete positivity of Ppx1, ̈ ̈ ̈,xL
Summary
In favour of motivating general problems, introducing general notions and stating our general theorems, we first go through concrete numerical examples that are special cases of our results. B. Generalizing [43], we show that codes with order-4 joint types (close to) Px1,x2,x3,x4 exist if and only if Px1,x2,x3,x4 is a completely positive tensor of order-4, i.e., Px1,x2,x3,x4 can be written as a convex combination of products of independent and identical distributions, k. One of the fundamental questions we are able to answer in this paper is the following: is it possible for us to design exponentially large codes such that no matter which codeword is transmitted and no matter how an adversary corrupts it via a legitimate action, the decoder is always able to output a list of at most (say). By setting the list size L 1 “ 1, results in [43] are recovered by our work
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