Efficient List-Decoding with Constant Alphabet and List Sizes

We present an explicit and efficient algebraic construction of capacity-achieving list decodable codes with both constant alphabet and constant list sizes. More specifically, for any R\in (0,1) and ϵ >0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/ϵ)O(1/ϵ 2), that can be list decoded from a (1-R-\ϵ)-fraction of errors with list size at most\exp (poly(1/ϵ)). Moreover, the codes can be encoded in time (1/\ϵ,n), the output list is contained in a linear subspace of dimension at most (1/\ϵ), and a basis for this subspace can be found in time (1/\ϵ, n). Thus, both encoding and list decoding can be performed in fully polynomial-time (1/\ϵ,n), except for pruning the subspace and outputting the final list which takes time\exp ((1/\ϵ)) (n). In contrast, prior explicit and efficient constructions of capacity-achieving list decodable codes either required a much higher complexity in terms of 1/\ϵ (and were additionally much less structured), or had super-constant alphabet or list sizes. Our codes are quite natural and structured. Specifically, we use algebraic-geometric (AG) codes with evaluation points restricted to a subfield, and with the message space restricted to a (carefully chosen) linear subspace. Our main observation is that the output list of AG codes with subfield evaluation points is contained in an affine shift of the image of a block-triangular-Toeplitz (BTT) matrix, and that the list size can potentially be reduced to a constant by restricting the message space to a BTT evasive subspace, which is a large subspace that intersects the image of any BTT matrix in a constant number of points. We further show how to explicitly construct such BTT evasive subspaces, based on the explicit subspace designs of Guruswami and Kopparty (Combinatorica, 2016), and composition.