文章基本信息

标题：Efficient List-Decoding with Constant Alphabet and List Sizes
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作者：Zeyu Guo ; Noga Ron-Zewi
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2020
卷号：2020
页码：1-38
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
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 ∈ (0, 1) and > 0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/) O(1/2 ) , that can be list decoded from a (1 − R − )-fraction of errors with list size at most exp(poly(1/)). Moreover, the codes can be encoded in time poly(1/, n), the output list is contained in a linear subspace of dimension at most poly(1/), and a basis for this subspace can be found in time poly(1/, n). Thus, both encoding and list decoding can be performed in fully polynomial-time poly(1/, n), except for pruning the subspace and outputting the final list which takes time exp(poly(1/))·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/ (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.