DeepAI AI Chat
Log In Sign Up

List Decoding of Direct Sum Codes

by   Vedat Levi Alev, et al.

We consider families of codes obtained by "lifting" a base code 𝒞 through operations such as k-XOR applied to "local views" of codewords of 𝒞, according to a suitable k-uniform hypergraph. The k-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 k 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 product, we also recover the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given word, as finding the optimal solution of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code 𝒞. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the "condition and round" approach used in several SOS algorithms can then be used to recover the required list of codewords.


page 1

page 2

page 3

page 4


List Decoding of Quaternary Codes in the Lee Metric

We present a list decoding algorithm for quaternary negacyclic codes ove...

Bounds on List Decoding of Linearized Reed-Solomon Codes

Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfi...

Block turbo decoding with ORBGRAND

Guessing Random Additive Noise Decoding (GRAND) is a family of universal...

Randomly punctured Reed–Solomon codes achieve list-decoding capacity over linear-sized fields

Reed–Solomon codes are a classic family of error-correcting codes consis...

Unique Decoding of Explicit ε-balanced Codes Near the Gilbert-Varshamov Bound

The Gilbert-Varshamov bound (non-constructively) establishes the existen...

Approximating Constraint Satisfaction Problems on High-Dimensional Expanders

We consider the problem of approximately solving constraint satisfaction...

List decoding of Convolutional Codes over integer residue rings

A convolutional code over [D] is a [D]-submodule of [D] where [D] stand...