In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes. Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under affine-transformations of the coordinate space. Lifting takes codes defined over a vector space of small dimension and lifts them to higher dimensions by requiring their restriction to every subspace of the original dimension to be a codeword of the code being lifted. While the operation is of interest on its own, this work focusses on new ranges of parameters that can be obtained by such codes, in the context of local correction and testing. In particular we present four interesting ranges of parameters that can be achieved by such lifts, all of which are new in the context of affine-invariance and some may be new even in general. The main highlight is a construction of high-rate codes with sublinear time decoding. The only prior construction of such codes is due to Kopparty, Saraf and Yekhanin [33]. All our codes are extremely simple, being just lifts of various parity check codes (codes with one symbol of redundancy), and in the final case, the lift of a Reed-Solomon code.We also present a simple connection between certain lifted codes and lower bounds on the size of "Nikodym sets". Roughly, a Nikodym set in F m q is a set S with the property that every point has a line passing through it which is almost entirely contained in S. While previous lower bounds on Nikodym sets were roughly growing as q m /2 m , we use our lifted codes to prove a lower bound of (1 − o(1))q m for fields of constant characteristic.

We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction.1. We show that every Kakeya set (a set of points that contains a line in every direction) in F n q must be of size at least q n /2 n . This bound is tight to within a 2 + o(1) factor for every n as q → ∞, compared to previous bounds that were off by exponential factors in n.2. We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input Λ (possibly correlated) random variables in {0, 1} N and a short random seed and output a single random variable in {0, 1} N that is statistically close to having entropy (1 − δ) · N when one of the Λ input variables is distributed uniformly. The seed we require is only (1/δ) · log Λ-bits long, which significantly improves upon previous construction of mergers. 3. Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting 1 − o(1) fraction of the min-entropy of the source. Previous results could extract only a constant fraction of the entropy while maintaining logarithmic seed length. * IAS. zeev.dvir@gmail.com.

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A subspace design is a collection {H 1 , H 2 , . . . , H M } of subspaces of F m q with the property that no low-dimensional subspace W of F m q intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal rate list-decodable codes over constant-sized large alphabets and sub-logarithmic (and even smaller) list size. Subspace designs are the only non-explicit part of their construction. In this paper, we give explicit constructions of subspace designs with parameters close to the probabilistic construction, and this implies the first deterministic polynomial time construction of list-decodable codes achieving the above parameters.Our constructions of subspace designs are natural and easily described, and are based on univariate polynomials over finite fields. Curiously, the constructions are very closely related to certain good listdecodable codes (folded RS codes and univariate multiplicity codes). The proof of the subspace design property uses the polynomial method (with multiplicities): Given a target low-dimensional subspace W , we construct a nonzero low-degree polynomial P W that has several roots for each H i that non-trivially intersects W . The construction of P W is based on the classical Wronskian determinant and the folded Wronskian determinant, the latter being a recently studied notion that we make explicit in this paper. Our analysis reveals some new phenomena about the zeroes of univariate polynomials, namely that polynomials with many structured roots or many high multiplicity roots tend to be linearly independent.

The list-decodability of random linear codes is shown to be as good as that of general random codes. Specifically, for every fixed finite field F q , p ∈ (0, 1 − 1/q) and ε > 0, it is proved that with high probability a random linear code C in F n q of rate (1 − H q (p) − ε) can be list decoded from a fraction p of errors with lists of size at most O(1/ε).This also answers a basic open question concerning the existence of highly list-decodable linear codes, showing that a list-size of O(1/ε) suffices to have rate within ε of the informationtheoretically optimal rate of 1 − H q (p). The best previously known list-size bound was q O(1/ε) (except in the q = 2 case where a list-size bound of O(1/ε) was known).The main technical ingredient in the proof is a strong upper bound on the probability that random vectors chosen from a Hamming ball centered at the origin have too many (more than Ω( )) vectors from their linear span also belong to the ball.