High Dimensional Expanders Imply Agreement Expanders

Abstract

We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC).Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are functions on a subset of F n whose restrictions to a prescribed set of affine lines has low degree. Alternatively, they are Tanner codes on high dimensional expanders, where the coordinates of the codeword correspond to triangles of a 2-dimensional expander, such that around every edge the local view forms a Reed-Solomon codeword.For some range of parameters our codes are provably locally testable, and their dimension is some fixed power of the block length. For another range of parameters our codes have distance and dimension that are both linear in the block length, but we do not know if they are locally testable. The codes also have the multiplication property: the coordinate-wise product of two codewords is a codeword in a related code.The definition of the codes relies on the construction of a specific family of simplicial complexes which is a slight variant on the coset complexes of Kaufman and Oppenheim. We show a novel way to embed the triangles of these complexes into F n , with the property that links of edges embed as affine lines in F n .We rely on this embedding to lower bound the rate of these codes in a way that avoids constraint-counting and thereby achieves non-trivial rate even when the local codes themselves have arbitrarily small rate, and in particular below 1/2.

Agreement Testing Theorems on Layered Set Systems

“…Building on [DS14], the main theorem in [DK17] shows that the √ d-dimensional skeleton S = X( √ d), of a d-dimensional two-sided high-dimensional expander, supports a c-sound agreement test for some constant c > 0. This gave the first agreement test on a sparse system of sets, that is, such that every vertex is contained in a constant number of sets.…”

“…That such VASA distributions are available is proven through a new type of random walk which we call the complement random walk, and is discussed separately below. The only previous work that analyzed an agreement test on a sparse set system (where this "large diameter" problem appears) was in [DK17]. Their solution circumvented this problem by reducing to the dense case in a certain way.…”

“…Several previous works [KM17,DK17,KO18b] analyzed random walks on high dimensional expanders 2 . In this work we study a new type of random walk which we call the complement random walk.…”

“…The VASA distribution is to choose s according to the X distribution and in it a, a , v uniformly so that they are all disjoint. Agreement tests for this example were analyzed in [DK17] for certain complexes X and certain bounds on the dimension k.…”

Tight Running Time Lower Bounds for Strong Inapproximability of Maximum k-Coverage, Unique Set Cover and Related Problems (via t-Wise Agreement Testing Theorem)

“…Hence, our agreement testing theorem is more closely related to the question of direct product test [GS00, DR06, IKW12, DS14b, DN17, GK18], for which the collection 7 S contains all subsets of sizes ℓ of the universe U and the local functions can be arbitrarily. There are also combinatorial agreement theorems beyond direct product tests; for instance, [DK17,DD19] consider collections S that corresponds to layered-set systems, such as high-dimensional expanders, and [DFH19] proves a "higher-dimensional" version of the direct product test.…”