Near-linear time decoding of Ta-Shma’s codes via splittable regularity
Autor: | Shashank Srivastava, Madhur Tulsiani, Fernando Granha Jeronimo |
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Rok vydání: | 2021 |
Předmět: |
Semidefinite programming
Hierarchy (mathematics) List decoding 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology Constraint satisfaction 01 natural sciences Combinatorics 010201 computation theory & mathematics Bounded function 0202 electrical engineering electronic engineering information engineering Binary code Time complexity Decoding methods Mathematics |
Zdroj: | STOC |
DOI: | 10.1145/3406325.3451126 |
Popis: | The Gilbert–Varshamov bound non-constructively establishes the existence of binary codes of distance 1/2−є/2 and rate Ω(є2). In a breakthrough result, Ta-Shma [STOC 2017] constructed the first explicit family of nearly optimal binary codes with distance 1/2−є/2 and rate Ω(є2+α), where α → 0 as є → 0. Moreover, the codes in Ta-Shma’s construction are є-balanced, where the distance between distinct codewords is not only bounded from below by 1/2−є/2, but also from above by 1/2+є/2. Polynomial time decoding algorithms for (a slight modification of) Ta-Shma’s codes appeared in [FOCS 2020], and were based on the Sum-of-Squares (SoS) semidefinite programming hierarchy. The running times for these algorithms were of the form NOα(1) for unique decoding, and NOє,α(1) for the setting of “gentle list decoding”, with large exponents of N even when α is a fixed constant. We derive new algorithms for both these tasks, running in time Oє(N). Our algorithms also apply to the general setting of decoding direct-sum codes. Our algorithms follow from new structural and algorithmic results for collections of k-tuples (ordered hypergraphs) possessing a “structured expansion” property, which we call splittability. This property was previously identified and used in the analysis of SoS-based decoding and constraint satisfaction algorithms, and is also known to be satisfied by Ta-Shma’s code construction. We obtain a new weak regularity decomposition for (possibly sparse) splittable collections W ⊆ [n]k, similar to the regularity decomposition for dense structures by Frieze and Kannan [FOCS 1996]. These decompositions are also computable in near-linear time O(|W |), and form a key component of our algorithmic results. |
Databáze: | OpenAIRE |
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