具有恒定字母表和列表大小的高效列表解码
Efficient List-Decoding with Constant Alphabet and List Sizes
摘要：
我们提出了一个明确的和有效的代数结构的能力实现列表可解码的代码与恒定的字母表和恒定的列表大小。更具体地说，对于任何$R \in (0,1)$和$ε>0$，我们给出了一个代数结构的一个无限家庭的纠错码率$R$，在一个字母表的大小$(1/ε)^{O(1/ε^2)}$，可以列表解码从$(1-R-ε)$ -分数的错误列表大小最多$\exp(\mathrm{poly}(1/ε))$。此外，可以在时间$\mathrm{poly}(1/ε, n)$中对代码进行编码，输出列表包含在维度至多为$\mathrm{poly}(1/ε)$的线性子空间中，并且可以在时间$\mathrm{poly}(1/ε, n)$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的$\mathrm{poly}(1/ε, n)$中执行，除了修剪子空间和输出最终列表需要时间$\exp(\mathrm{poly}(1/ε))\cdot\mathrm{poly}(n)$。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）代码的评价点限制在一个子域，并与消息空间限制在一个（精心选择的）线性子空间。我们的主要观察结果是，输出列表的AG代码与子域评估点包含在仿射移位的图像的块三角Toeplitz（BTT）矩阵，列表的大小可以潜在地减少到一个常数，通过限制消息空间的BTT回避子空间，这是一个大的子空间，相交的图像中的任何BTT矩阵在一个常数数量的点。我们进一步展示了如何基于Guruswami和Kopparty的显式子空间设计（Combinatorica，2016）和合成显式构造这样的BTT回避子空间。
英文摘要：
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 $ε>0$, we give an algebraic construction of an infinite family of error-correcting codes of rate $R$, over an alphabet of size $(1/ε)^{O(1/ε^2)}$, that can be list decoded from a $(1-R-ε)$-fraction of errors with list size at most $\exp(\mathrm{poly}(1/ε))$. Moreover, the codes can be encoded in time $\mathrm{poly}(1/ε, n)$, the output list is contained in a linear subspace of dimension at most $\mathrm{poly}(1/ε)$, and a basis for this subspace can be found in time $\mathrm{poly}(1/ε, n)$. Thus, both encoding and list decoding can be performed in fully polynomial-time $\mathrm{poly}(1/ε, n)$, except for pruning the subspace and outputting the final list which takes time $\exp(\mathrm{poly}(1/ε))\cdot\mathrm{poly}(n)$. 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.
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