  We consider families of codes obtained by "lifting" a base code $\mathcal{C}$
through operations such as $k$-XOR applied to "local views" of codewords of
$\mathcal{C}$, 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 $\mathcal{C}$. 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.
