List decoding of direct sum codes

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 indeed satisfied by the collection of length k walks on a sufficiently strong expanding graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings corresponding to the above hypergraph families. Using known connections between direct sum and direct product, we also recover (and strengthen) 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 (possibly corrupted) word, as finding the optimal solution to an instance 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 by some of the authors 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.