The Direct Product encoding of a string @math on an underlying domain @math , is a function DP @math which gets as input a set @math and outputs @math restricted to @math . In the Direct Product Testing Problem, we are given a function @math , and our goal is to test whether @math is close to a direct product encoding, i.e., whether there exists some @math such that on most sets @math , we have @math DP @math . A natural test is as follows: select a pair @math according to some underlying distribution over @math , query @math on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set @math and is referred to as a test graph. The testability of direct products was studied over various specific domains and test graphs (for example see Dinur-Steurer [CCC'14]; Dinur-Kaufman [FOCS'17]). In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem? Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every @math and @math we provide a direct product domain @math of size @math , called the Sliding Window domain for which we prove direct product testability.
@cite_9: A direct product function is a function of the form g(x_1, ldots, x_k)=(g_1(x_1), ldots, g_k(x_k)). We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability. This local testing question comes up naturally in the context of PCPs, where direct products play a prominent role for gap amplification. We consider the following natural two query test for a given function f:[N]^kto[M]^k begin quote bf Two query direct product test : Choose x, y that agree on a random set A of t coordinates and accept if f(x)_A=f(y)_A. End quote We provide a comprehensive analysis of this test for all parameters N, M, k, tle O(k) and success probability delta>0. Our main result is that if a given function f:[N]^kto[M]^k passes the test with probability delta ge 1-eps then there is a direct product function g such that Pr[f(x)=g(x)]ge 1-O(eps). This is the first result relating success in the above (or any) test to the fraction of the domain on which f is equal to a direct product function. This test has been analyzed in previous works for the case t ll k ll N, and results show closeness of f to a direct product under a less natural measure of "approximate agreement". In the small soundness regime, we prove that if the test above passes with probability delta ge exp(-k), then the function agrees with a direct product function on local parts of the domain. This extends the previous range of parameters of delta ge exp(-sqrt[3]k) to the entire meaningful range of delta>exp(-k).
@cite_11: We develop the notion of "double samplers", first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders. We show how double samplers give a generic way of amplifying distance in a way that enables efficient list-decoding. There are many error correcting code constructions that achieve large distance by starting with a base code @math with moderate distance, and then amplifying the distance using a sampler, e.g., the ABNNR code construction [IEEE Trans. Inform. Theory, 38(2):509--516, 1992.]. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm and the list decoding algorithm is oblivious to the base code @math (i.e., it runs the unique decoder for @math in a black box way). Our list-decoding algorithm works as follows: it uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.
@cite_6: The @math -fold direct sum encoding of a string @math is a function @math that takes as input sets @math of size @math and outputs @math . In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form @math for some @math and rejects with probability @math functions @math that are @math -far from being a direct sum encoding, where the constant behind the @math notation is independent of @math . This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of @math with vectors in @math in the natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the @math th layer of the hypercube (i.e., the set of @math -bit strings with Hamming weight @math ). Tensor power testing: By moving to @math notation, the direct sum encoding is equivalent (up to a difference thatis negligi...
@cite_2: 
The state of the art in this context is the result of Dinur and Steurer The result in @cite_9 is stated in the language of tuples, i.e., the domain is a subset of @math , but their result also holds when the domain is a collection of @math -sized subsets of @math . See @cite_6 for more details. @cite_9 dealing with the domain @math where @math varies between @math and @math . They analyze the aforementioned two-query test with @math -intersection size. They analyze it in the high acceptance regime and show that @math indeed admits a direct product testing theorem. The proof is quite involved and in particular analyzes first the low acceptance regime. Recently, in a breakthrough paper, Dinur and Kaufman @cite_2 analyzed the two-query test when the underlying domain is obtained from the set of faces of a Ramanujan complex. Their approach crucially relies on the result of @cite_9 . More recently @cite_11 introduced the notion of and remarked that it might admit a direct product theorem.