We prove that if a linear error correcting code
$\C:\{0,1\}^n\to\{0,1\}^m$ is such that a bit of the message can
be probabilistically reconstructed by looking at two entries of a
corrupted codeword, then $m = 2^{\Omega(n)}$. We also present
several extensions of this result.

We show a reduction from the ... more >>>

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

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Many of the keystream generators which are used in practice are LFSR-based in the sense
that they produce the keystream according to a rule $y=C(L(x))$,
where $L(x)$ denotes an internal linear bitstream, produced by a small number of parallel
linear feedback shift registers (LFSRs),
and $C$ denotes ... more >>>