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.

中文翻译：

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具有恒定字母和列表大小的高效列表解码

我们提出了具有恒定字母表和恒定列表大小的容量实现列表可解码代码的明确和有效的代数构造。更具体地说，对于任何 $R \in (0,1)$ 和 $\epsilon>0$，我们在大小为 $( 1/\epsilon)^{O(1/\epsilon^2)}$，可以从列表大小至多 $\exp(\ mathrm{poly}(1/\epsilon))$。此外，代码可以在时间 $\mathrm{poly}(1/\epsilon, n)$ 中进行编码，输出列表包含在最多维度为 $\mathrm{poly}(1/\epsilon) 的线性子空间中$，并且可以在时间 $\mathrm{poly}(1/\epsilon, n)$ 中找到该子空间的基。因此，编码和列表解码都可以在完全多项式时间内执行 $\mathrm{poly}(1/\epsilon, n)$，除了修剪子空间和输出最终列表需要时间 $\exp(\mathrm{poly }(1/\epsilon))\cdot\mathrm{poly}(n)$。我们的代码非常自然和结构化。具体来说，我们使用代数几何 (AG) 代码，其评估点仅限于子域，消息空间仅限于（精心选择的）线性子空间。我们的主要观察结果是，具有子场评估点的 AG 代码的输出列表包含在块三角形托普利茨 (BTT) 矩阵的图像的仿射移位中，并且列表大小可以通过限制消息空间到 BTT 回避子空间，这是一个很大的子空间，它与任何 BTT 矩阵的图像相交于恒定点数。我们进一步展示了如何基于 Guruswami 和 Kopparty (Combinatorica, 2016) 的显式子空间设计和组合来显式构建此类 BTT 回避子空间。