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.

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We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string ... more >>>