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].

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We consider the problem of approximately solving constraint satisfaction problems with arity k > 2 (k-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of k-CSPs, which are also highly expanding, are wellknown to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander.We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter γ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet [q] is a highdimensional expander with parameter γ, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error ε using q O(k) Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank r for a threshold τ = (ε/k) O(1) · (1/q) O(k) , then it is possible to approximately solve the instance up to additive error ε, using r · q O(k) · (k/ε) O(1) levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small γ) have threshold rank 1 according to our definition. spectral expander 1 . Here, the graph G(X a ) is defined to have the vertex set {i | a ∪ {i} ∈ X} and edge-set {i, j | a ∪ {i, j} ∈ X}. If the (normalized) second singular value of each of the neighborhood graphs is bounded by γ, X is said to be a γ-high-dimensional expander (γ-HDX).Note that (the downward closure of) a random sparse (d + 1)-uniform hypergraph, say with n vertices and c · n edges, is very unlikely to be a d-dimensional expander. With high probability, no two hyperedges share more than one vertex and thus for any i ∈ [n], the neighborhood graph G i is simply a disjoint union of cliques of size d, which is very far from an expander. While random hypergraphs do not yield high-dimensional expanders, such objects are indeed known to exists via (surprising) algebraic constructions [LSV05b, LSV05a, KO18a, CTZ18] and are known to have several interesting properties and applications [KKL16, DHK + 18, KM17, KO18b, DDFH18, DK17, PRT16].Expander graphs can simply be thought of as the one-dimensional case of the above definition. The results of Barak, Raghavendra and Steurer [BRS11] for 2-CSPs yield that if the constraint graph of a 2-CSP instance (with size n and alphabet size q) is a sufficiently good (one dimensional) spectral expander, then one can efficiently find solutions satisfying OPT − ε fraction of constraints, where OPT denotes the maximum fraction of ...

We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semidefinite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following results:-We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] andZémor [IEEE Trans. Inf. Theory 2001] with alphabet size q, block-length n and distance δ, based on an expander graph with degree d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.-We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code.In particular, starting from any base code unique-decodable up to distance δ, one can obtain near-MDS codes with rate R and distance 1 − R − ε, list-decodable up to the Johnson bound in time n O ε,δ (1) .-We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size q, block-length n and distance δ based on a square Cayley complex with generator sets of size d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.

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