  Baranyai's theorem is a well-known theorem in the theory of hypergraphs. A
corollary of this theorem says that one can partition the family of all
$u$-subsets of an $n$-element set into ${n-1\choose u-1}$ sub-families such
that each sub-family form a partition of the $n$-element set, where $n$ is
divisible by $u$. In this paper, we present a coding-theoretic application of
Baranyai's theorem (or equivalently, the corollary). More precisely, we propose
the first purely combinatorial construction of locally decodable codes. Locally
decodable codes are error-correcting codes that allow the recovery of any
message bit by looking at only a few bits of the codeword. Such codes have
attracted a lot of attention in recent years. We stress that our construction
does not improve the parameters of known constructions. What makes it
interesting is the underlying combinatorial techniques and their potential in
future applications.
