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 is an element of (0, 1) and epsilon > 0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/epsilon)(O(1/epsilon 2)), that can be list decoded from a (1 - R - epsilon)-fraction of errors with list size at most exp(poly(1/epsilon)). Moreover, the codes can be encoded in time poly(1/epsilon, n), the output list is contained in a linear subspace of dimension at most poly(1/epsilon), and a basis for this subspace can be found in time poly(1/epsilon, n). Thus, both encoding and list decoding can be performed in fully polynomial-time poly(1/epsilon, n), except for pruning the subspace and outputting the final list which takes time exp(poly(1/epsilon)) . poly(n). In contrast, prior explicit and efficient constructions of capacity-achieving list decodable codes either required a much higher complexity in terms of 1/epsilon (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.

Automated summary

The paper presents an explicit and efficient algebraic construction for capacity-achieving list decodable codes. The proposed method uses algebraic-geometric codes with restricted evaluation points and message space, allowing for polynomial-time encoding and decoding with reduced list size.