  We study the following combinatorial version of the Slepian-Wolf coding
scheme. Two isolated Senders are given binary strings $X$ and $Y$ respectively;
the length of each string is equal to $n$, and the Hamming distance between the
strings is at most $\alpha n$. The Senders compress their strings and
communicate the results to the Receiver. Then the Receiver must reconstruct
both strings $X$ and $Y$. The aim is to minimise the lengths of the transmitted
messages.
  For an asymmetric variant of this problem (where one of the Senders transmits
the input string to the Receiver without compression) with deterministic
encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany.
In our paper we prove a new lower bound for the schemes with syndrome coding,
where at least one of the Senders uses linear encoding of the input string.
  For the combinatorial Slepian-Wolf problem with randomized encoding the
theoretical optimum of communication complexity was recently found by the first
author, though effective protocols with optimal lengths of messages remained
unknown. We close this gap and present a polynomial time randomized protocol
that achieves the optimal communication complexity.
