A Coding-Theoretic Application of Baranyai's Theorem

Abstract

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.