Efficient List-Decoding with Constant Alphabet and List Sizes

TL;DR

This paper introduces a new algebraic construction of capacity-achieving list-decodable codes with constant alphabet and list sizes, enabling efficient encoding and decoding within polynomial time, and uses structured algebraic-geometric codes with specialized subspaces.

Contribution

It provides an explicit, algebraic construction of list-decodable codes with constant alphabet and list sizes, achieving capacity with efficient algorithms and novel subspace techniques.

Findings

Codes achieve capacity with constant alphabet and list size

Encoding and decoding are polynomial-time algorithms

Subspace constructions reduce list size to a constant

Abstract

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 $\exp(\mathrm{poly}(1/\epsilon))\cdot\mathrm{poly}(n)$. 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.