  Codes on hypergraphs are an extension of the well-studied family of codes on
bipartite graphs. Bilu and Hoory (2004) constructed an explicit family of codes
on regular t-partite hypergraphs whose minimum distance improves earlier
estimates of the distance of bipartite-graph codes. They also suggested a
decoding algorithm for such codes and estimated its error-correcting
capability.
  In this paper we study two aspects of hypergraph codes. First, we compute the
weight enumerators of several ensembles of such codes, establishing conditions
under which they attain the Gilbert-Varshamov bound and deriving estimates of
their distance. In particular, we show that this bound is attained by codes
constructed on a fixed bipartite graph with a large spectral gap.
  We also suggest a new decoding algorithm of hypergraph codes that corrects a
constant fraction of errors, improving upon the algorithm of Bilu and Hoory.
