List Decoding of Direct Sum Codes
Vedat Levi Alev, Fernando Granha Jeronimo, Dylan Quintana and, Shashank Srivastava, Madhur Tulsiani

TL;DR
This paper develops a general framework for list decoding of lifted codes obtained via hypergraph-based operations, leveraging Sum-of-Squares SDP relaxations and expansion properties to recover codewords efficiently.
Contribution
It introduces a novel decoding framework for lifted codes using SOS relaxations and hypergraph expansion, extending previous results to broader code families.
Findings
Framework applies to hypergraphs from expanders and high-dimensional expanders.
List decoding algorithms are derived from CSP solving techniques.
Connections to direct product codes are established.
Abstract
We consider families of codes obtained by "lifting" a base code through operations such as -XOR applied to "local views" of codewords of , according to a suitable -uniform hypergraph. The -XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the collection of length walks on an expander graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings on the above hypergraph families. Using known connections between direct sum and direct…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Algorithms and Data Compression · Error Correcting Code Techniques
