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
$\epsilon >0$
, 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
$\exp (\mathrm {poly}(1/ \epsilon))$
. Moreover, the codes can be encoded in time
$\mathrm {poly}(1/ \epsilon,n)$
, the output list is contained in a linear subspace of dimension at most
$\mathrm {poly}(1/ \epsilon)$
, and a basis for this subspace can be found in time
$\mathrm {poly}(1 /\epsilon, n)$
. Thus, both encoding and list decoding can be performed in
fully polynomial-time

$\mathrm {poly}(1/ \epsilon,n)$
, except for pruning the subspace and outputting the final list which takes time
$\exp (\mathrm {poly}(1/ \epsilon)) \cdot \mathrm {poly} (n)$
. 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.