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.

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs, and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs isΘ(where n is the number of vertices in the graph (andΘ(f (n)) means Θ(f (n) · polylog(f (n)))). Thus there is a large gap between the complexity of testing in the two cases. In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity isÕ(min( √ n, n 2 /m)) where m is the number of edges in the graph, and match it with an almost tight lower bound.

A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector's bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n = 2 m is a word in the r-th order Reed-Muller code R(r, m) of length n = 2 m . For a given integer r ≥ 1, and real > 0, the algorithm queries the input vector v at O( 1 + r2 2r ) positions. On one hand, if v is at distance at least n from the closest codeword, then the algorithm discovers it with probability at least 2/3. On the other hand, if v is a codeword, then it always passes the test. Our result is almost tight: any algorithm for testing R(r, m) must perform Ω( 1 + 2 r ) queries.

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman [4] on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for all distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura [8] on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to all distances.

A degree-d polynomial p in n variables over a field F is equidistributed if it takes on each of its |F| values close to equally often, and biased otherwise. We say that p has a low rank if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [GT07] have shown that bias imply low rank over large fields (i.e. for the case d < |F|). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be approximated by few polynomials of bounded degree, can be computed by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.