A Coding-Theoretic Application of Baranyai’s Theorem

Baranyai´s theorem is well known 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;u-1) subfamilies such that each subfamily forms 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. More precisely, we propose a combinatorial construction of locally decodable codes. Locally decodable codes are error-correcting codes that allow the recovery of any message symbol by looking at only a few symbols of the codeword. The number of looked codeword symbols is called query complexity. Such codes have attracted a lot of attention in recent years. The Walsh-Hadamard code is a well-known binary two-query locally decodable code of exponential length that can recover any message bit using 2 bits of the codeword. Our construction can give locally decodable codes over small finite fields for any constant query complexities. In particular, it gives a ternary two-query locally decodable code of length asymptotically shorter than the Walsh-Hadamard code.