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 sizes. More specifically, for any $R \in (0,1)$ notation="LaTeX">$\epsilon >0$ , we give infinite family error-correcting rate notation="LaTeX">$R$ over size notation="LaTeX">$(1/\epsilon)^{O(1/\epsilon ^{2})}$ that can be decoded from a notation="LaTeX">$(1-R-\epsilon)$ -fraction errors at most notation="LaTeX">$\exp (\mathrm {poly}(1/ \epsilon))$ . Moreover, the encoded in time notation="LaTeX">$\mathrm \epsilon,n)$ output is contained linear subspace dimension \epsilon)$ basis this found {poly}(1 /\epsilon, n)$ Thus, both encoding decoding performed xmlns:xlink="http://www.w3.org/1999/xlink">fully polynomial-time except pruning outputting final which takes \epsilon)) \cdot \mathrm {poly} (n)$ In contrast, prior constructions either required much higher complexity terms notation="LaTeX">$1/ \epsilon $ (and were additionally less structured), or had super-constant Our are quite natural structured. Specifically, use algebraic-geometric (AG) evaluation points restricted to subfield, message space (carefully chosen) subspace. main observation AG subfield affine shift image xmlns:xlink="http://www.w3.org/1999/xlink">block-triangular-Toeplitz (BTT) xmlns:xlink="http://www.w3.org/1999/xlink">matrix potentially reduced by restricting BTT xmlns:xlink="http://www.w3.org/1999/xlink">evasive subspace large intersects matrix number points. further show how explicitly construct such evasive subspaces, based on designs Guruswami Kopparty ( xmlns:xlink="http://www.w3.org/1999/xlink">Combinatorica 2016), composition.