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 e (0,1) and ?>0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/?)O(1/?2), that can be list decoded from a (1-R-?)-fraction of errors with list size at most exp(poly(1/?)). Moreover, the codes can be encoded in time poly(1/?, n), the output list is contained in a linear subspace of dimension at most poly(1/?), and a basis for this subspace can be found in time poly(1/?, n). Thus, both encoding and list decoding can be performed in fully polynomial-time poly(1/?, n), except for pruning the subspace and outputting the final list which takes time exp(poly(1/?)) · 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/? (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.
|Title of host publication||STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing|
|Editors||Samir Khuller, Virginia Vassilevska Williams|
|Publisher||Association for Computing Machinery|
|Number of pages||14|
|State||Published - 15 Jun 2021|
|Event||53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy|
Duration: 21 Jun 2021 → 25 Jun 2021
|Name||Proceedings of the Annual ACM Symposium on Theory of Computing|
|Conference||53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021|
|Period||21/06/21 → 25/06/21|
Bibliographical noteFunding Information:
∗Research supported in part by ISF grant 735/20.
© 2021 ACM.
- algebraic codes
- algebraic-geometric codes
- error-correcting codes
- explicit constructions
- list decoding
ASJC Scopus subject areas