On the List-Decodability of Random Linear Codes

For every fixed finite field \F_q, $p \in (0,1-1/q)$ and $\epsilon > 0$, we prove that with high probability a random subspace $C$ of \F_q^n of dimension $(1-H_q(p)-\epsilon)n$ has the property that every Hamming ball of radius $pn$ has at most $O(1/\epsilon)$ codewords. This answers a basic open question concerning the list-decodability of linear codes, showing that a list size of $O(1/\epsilon)$ suffices to have rate within $\epsilon$ of the "capacity" $1-H_q(p)$. Our result matches up to constant factors the list-size achieved by general random codes, and gives an exponential improvement over the best previously known list-size bound of $q^{O(1/\epsilon)}$. The main technical ingredient in our proof is a strong upper bound on the probability that $\ell$ random vectors chosen from a Hamming ball centered at the origin have too many (more than $\Theta(\ell)$) vectors from their linear span also belong to the ball.Comment: 15 page

Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on $n$-vertex hyper-graphs: * Coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, polylog n colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a 'query doubling' method. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form.Comment: 25 page

Dual Vectors and Lower Bounds for the Nearest Lattice Point Problem

Abstract: We prove that given a point ~z outside a given lattice L then there is a dual vector which gives a fairly good estimate for how far from the lattice the vector is. To S be more precise, there is a set of translated hyperplanes Hi such that L i Hi and d(~z � S i Hi

Solving simultaneous modular equations of low degree

Abstract: We consider the problem of solving systems of equations Pi(x) 0 (mod ni) i = 1:::k where Pi are polynomials of degree d and the ni are distinct relatively prime numbers and x < min(ni). We prove that if k> d(d+1) we can recover x in polynomial 2 time provided min(ni)> 2d2. As a consequence the RSA cryptosystem used with a small exponent is not a good choice to use as a public key cryptosystem in a large network. We also show that a protocol by Broder and Dolev [4] is insecure if RSA with a small exponent is used. Warning: Essentially this paper has been published in SIAM Journal on Computing and is hence subject to copyright restrictions. It is for personal use only. 1