For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $\varepsilon >
0$, we prove that with high probability a random subspace $C$ of
$\F_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the
property that every Hamming ball of radius $pn$ has at most
$O(1/\varepsilon)$ codewords.

This ... more >>>

We show that any explicit example for a tensor $A:[n]^r \rightarrow \mathbb{F}$ with tensor-rank
$\geq n^{r \cdot (1- o(1))}$,
(where $r = r(n) \leq \log n / \log \log n$), implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over $\mathbb{F}$. This shows that strong enough ... more >>>

We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, $Sam_d$ where $d$ is the recursion depth, is based on ... more >>>