An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the c^3-problem.

We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness. This paper presents the second installment in a line of work by various subsets of the authors (with additional contributions by Barak, Kothari, and Steurer (ITCS'19)), which led to a proof of the 2-to-2 Games Conjecture.

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 $\mathbb{F}^n$ 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 $\mathbb{F}^n$, with the property that links of edges embed as affine lines in $\mathbb{F}^n$. 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 tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most PCP constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the main question is whether average agreement of the local pieces implies that there exists a global function that agrees with most local restrictions. There are very few structures that support agreement tests, essentially either coming from algebraic low degree tests or from direct product tests (and recently also from high-dimensional expanders). In this work, we prove a new agreement theorem which extends direct product tests to higher dimensions, analogous to how low degree tests extend linearity testing. As a corollary of our main theorem, it follows that an ensemble of small graphs on overlapping sets of vertices can be glued together to one global graph assuming they agree with each other on average. We prove the agreement theorem by (re)proving the agreement theorem for dimension 1, and then generalizing it to higher dimensions (with the dimension 1 case being the direct product test, and dimension 2 being the graph case). A key technical step in our proof is the reverse union bound, which allows us to treat dependent events as if they are disjoint, and may be of independent interest. An added benefit of the reverse union bound is that it can be used to show that the "majority decoded" function also serves as a global function that explains the local consistency of the agreement theorem, a fact that was not known even in the direct product setting (dimension 1) prior to our work.

Let X be a family of k-element subsets of [n] and let {f(s):s ->Sigma vertical bar s epsilon X} be an ensemble of local functions, each defined over a subset s subset of [n]. Is there a global function G:[n]->Sigma such that f(s) = G|(s) for all s epsilon X ? An agreement test is a randomized property tester for this question. One such test is the V-test, that chooses a random pair of sets s(1),s(2) epsilon X with prescribed intersection size and accepts if f(s1),f(s2) agree on the elements in s(1) boolean AND s(2). The low acceptance (or 1%) regime is concerned with the situation that the test succeeds with low but non-negligible probability Agree({f(s)}) >= epsilon > 0. A classical low acceptance agreement theorem says Agree ({f(s)})> epsilon double right arrow (*) $G:[n]->Sigma, P-s[f(s) approximate to (0.99) G|(s)] >= poly(epsilon). Such statements are motivated by PCP questions. The case X= ( [n] k) is well-studied and known as direct product testing, which is related to the parallel repetition theorem. Finding sparser families X that satisfy (*) is known as derandomized direct product testing. Prior to this work, the sparsest family satisfying (*) had |X| approximate to n(25), and we show X with |X| approximate to n(2). We study the general behavior of high dimensional expanders with respect to agreement tests in the low acceptance regime. High dimensional expanders, even very sparse ones with |X|=O(n), are known to satisfy the high acceptance variant (where epsilon =1-o(1)). It has been an open challenge to analyze the low acceptance regime. Surprisingly, topological covers of X play an important role. We show that: (1) If X has no connected covers, then (*) holds, provided that X satisfies an additional expansion property, called swap cosystolic expansion. (2) If X has a connected cover, then (*) fails. (3) If X has a connected cover (and swap-cosystolic-expansion), we replace (*) by a statement that takes covers into account: Agree ({f(s)}) > epsilon double right arrow (**) there exists cover rho : Y (SIC) X, and G : Y(0)->Sigma, such that P-s(SIC)s[f(s) approximate to (0.99) G|(s(SIC))] >= poly(epsilon), where s((SIC)) (SIC) s means that rho(s((SIC))) = s. The property of swap-cosystolic-expansion holds for quotients of the Bruhat Tits buildings. As a corollary we derive (*) for X being a spherical building, yielding a derandomized family with |X| approximate to n(2). We also derive (**) for LSV complexes X, for which |X|=O(n).

We give a structure theorem for Boolean functions on the $p$-biased hypercube which are $\epsilon$-close to degree $d$ in $L_2$, showing that they are close to sparse juntas. Our structure theorem implies that such functions are $O(\epsilon^{C_d} + p)$-close to constant functions. We pinpoint the exact value of the constant $C_d$. We also give an analogous result for monotone Boolean functions on the biased hypercube which are $\epsilon$-close to degree $d$ in $L_2$, showing that they are close to sparse DNFs. Our structure theorems are optimal in the following sense: for every $d,\epsilon,p$, we identify a class $\mathcal{F}_{d,\epsilon,p}$ of degree $d$ sparse juntas which are $O(\epsilon)$-close to Boolean (in the monotone case, width $d$ sparse DNFs) such that a Boolean function on the $p$-biased hypercube is $O(\epsilon)$-close to degree $d$ in $L_2$ iff it is $O(\epsilon)$-close to a function in $\mathcal{F}_{d,\epsilon,p}$.

We introduce a high-dimensional cubical complex, for any dimension t is an element of N, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t = 2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes h(1), ... , h(t) as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t = 4 our construction gives a new family of "almost-good" quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

Property testing has been a major area of research in computer science in the last three decades. By property testing we refer to an ensemble of problems, results and algorithms which enable to deduce global information about some data by only reading small random parts of it. In recent years, this theory found its way into group theory, mainly via group stability. In this paper, we study the following problem: Devise a randomized algorithm that given a subgroup H of G, decides whether H is the whole group or a proper subgroup, by checking whether a single (random) element of G is in H. The search for such an algorithm boils down to the following purely group theoretic problem: For G of rank k, find a small as possible test subset A⊆ G such that for every proper subgroup H, |H∩ A|≤ (1-δ)|A| for some absolute constant δ>0, which we call the detection probability of A. It turns out that the search for sets A of size linear in k and constant detection probability is a non-commutative analogue of the classical search for families of good error correcting codes. This paper is devoted to proving that such test subsets exist, which implies good universal error correcting codes exist – providing a far reaching generalization of the classical result of Shannon. In addition, we study this problem in certain subclasses of groups – such as abelian, nilpotent, and finite solvable groups – providing different constructions of test subsets for these subclasses with various qualities. Finally, this generalized theory of non-commutative error correcting codes suggests a plethora of interesting problems and research directions.

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to being implementable in practice, due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalises the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction, we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalises Kahale’s average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [N] and let {f(s) : s -> Sigma vertical bar s is an element of X} be an ensemble of local functions, each defined over a subset s subset of [N]. Suppose that we run the following so-called agreement test: choose a random pair of sets s(1), s(2) is an element of X that intersect on root k elements, and accept if f(s1), f(s2) agree on the elements in s(1) boolean AND s(2). We denote the success probability of this test by Agree({f(s)}). Given that Agree({f(s)}) = epsilon > 0, is there a global function G : [N] -> Sigma such that f(s) = G vertical bar(s) for a non-negligible fraction of s is an element of X ? We construct a family X of k-subsets of [N] such that vertical bar X vertical bar = O(N) and such that it satisfies the low acceptance agreement theorem. Namely, Agree({f(s)}) > epsilon double right arrow there exists G : [N] -> Sigma, P-s[f(s) (0.99)approximate to G vertical bar s] >= poly(epsilon). A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The latter serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of symplectic HDXs.

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only |X(k-1)|=O(n) points in contrast to ( [ n; k ]) points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

Given a family $X$ of subsets of $[n]$ and an ensemble of local functions $\{f_s:s\to\Sigma\; | \; s\in X\}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]\to\Sigma$ such that $f_s=G|_s$ for many sets $s$. A "classical" small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that \[\tag{$LD$} Agree(\{f_s\}) > \varepsilon \quad \Longrightarrow \quad \exists G^1,\dots, G^\ell,\quad P_s[f_s\overset{0.99}{\approx}G^i|_s]\geq poly(\varepsilon),\;i=1,\dots,\ell. \] Such a statement is motivated by PCP questions and has been shown in the case where $X=\binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: \[ \tag{$LFD$} Agree(\{f_s\})> \varepsilon \Longrightarrow \quad \exists\text{ cover }\rho:Y\twoheadrightarrow X,\text{ and }G:Y(0)\to\Sigma,\text{ such that }\] \[P_{{\tilde s\twoheadrightarrow s}}[f_s \overset{0.99}{\approx} G|_{\tilde s}] \geq poly(\varepsilon),\] where ${\tilde s\twoheadrightarrow s}$ means that $\rho(\tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.

We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain (2m× m)V   →δ0  (2m)E →δ1  2F where V (X-checks) are the vertices, E (qubits) are the edges, and F (Z-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes CA,CB:2m→2Δ where Δ is the regularity of the underlying Cayley graphs. One of the main ingredients in the analysis is a proof of an essentially-optimal robustness property for the tensor product of two random codes.

Given a function f : [N]k → [M]k, the Z-test is a three query test for checking if a function f is a direct product, namely if there are functions g1, ... gk : [N] → [M] such that f(x1, ..., xk) = (g1(x1), ... gk(xk)) for every input x e [N]k.This test was introduced by Impagliazzo et. al. (SICOMP 2012), who showed that if the test passes with probability e > [EQUATION] then f is Ω(e) close to a direct product function some precise sense. It remained an open question whether the soundness of this test can be pushed all the way down to exp(−k) (which would be optimal). This is our main result: we show that whenever f passes the Z test with probability e > exp(−k), there must be a global reason for this: namely, f must be close to a product function on some Ω(e) fraction of its domain.Towards proving our result we analyze the related (two-query) V-test, and prove a restricted global structure theorem for it. Such theorems were also proven previous works on direct product testing the small soundness regime. The most recent work, by Dinur and Steurer (CCC 2014), analyzed the V test the exponentially small soundness regime. We strengthen their conclusion of that theorem by moving from an in expectation statement to a stronger concentration of measure type of statement, which we prove using hyper-contractivity. This stronger statement allows us to proceed to analyze the Z test.We analyze two variants of direct product tests. One for functions on ordered tuples, as above, and another for functions on sets, [EQUATION]. The work of Impagliazzo et. al was actually focused only on functions of the latter type, i.e. on sets. We prove exponentially small soundness for the Z-test for both variants. Although the two appear very similar, the analysis for tuples is more tricky and requires some additional ideas.

We point out an error in the paper "Linear Time Encoding of LDPC Codes" (by Jin Lu and José M. F. Moura, IEEE Trans). The paper claims to present a linear time encoding algorithm for every LDPC code. We present a family of counterexamples, and point out where the analysis fails. The algorithm in the aforementioned paper fails to encode our counterexample, let alone in linear time.

A seminal result in learning theory characterizes the PAC learnability of binary classes through the Vapnik-Chervonenkis dimension. Extending this characterization to the general multiclass setting has been open since the pioneering works on multiclass PAC learning in the late 1980s. This work resolves this problem: we characterize multiclass PAC learnability through the DS dimension, a combinatorial dimension defined by Daniely and Shalev-Shwartz, (2014). The classical characterization of the binary case boils down to empirical risk minimization. In contrast, our characterization of the multiclass case involves a variety of algorithmic ideas; these include a natural setting we call list PAC learning. In the list learning setting, instead of predicting a single outcome for a given unseen input, the goal is to provide a short menu of predictions. Our second main result concerns the Natarajan dimension, which has been a central candidate for characterizing multiclass learnability. This dimension was introduced by Natarajan (1988) as a barrier for PAC learning. He furthered showed that it is the only barrier, provided that the number of labels is bounded. Whether the Natarajan dimension characterizes PAC learnability in general has been posed as an open question in several papers since. This work provides a negative answer: we construct a non-learnable class with Natarajan dimension 1. For the construction, we identify a fundamental connection between concept classes and topology (i.e., colorful simplicial complexes). We crucially rely on a deep and involved construction of hyperbolic pseudo-manifolds by Januszkiewicz and Światkowski. It is interesting that hyperbolicity is directly related to learning problems that are difficult to solve although no obvious barriers exist. This is another demonstration of the fruitful links machine learning has with different areas in mathematics.

Let X, Y be simplicial complexes and let $$f:Y \rightarrow X$$ be a simplicial surjective map. We introduce a notion of deficiency of f, denoted by $$m_f(Y)$$ , that measures the average local failure of $$f:Y \rightarrow X$$ to be a covering map. We show, roughly speaking, that if $$m_f(Y)$$ is small and and if the non-abelian cosystolic expansion of X is large, then f is close to a genuine covering map. Our main result is a lower bound on the 1-cosystolic expansion with G coefficients of geometric lattices, with an application to near coverings of the 2-dimensional spherical building $$A_{3}({\mathbb {F}}_q)$$ .