A code is called a q-query locally decodable code (LDC) if there is a
randomized decoding algorithm that, given an index i and a received word w
close to an encoding of a message x, outputs xi by querying only at most
q coordinates of w. Understanding the tradeoffs between the dimension,
length and query complexity of LDCs is a fascinating and unresolved research
challenge. In particular, for 3-query binary LDCs of dimension k and length
n, the best known bounds are: 2ko(1)≥n≥Ω~(k2).
In this work, we take a second look at binary 3-query LDCs. We investigate
a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query
LDCs. We prove an upper bound on the number of edges in these hypergraphs,
reproducing the known lower bound of Ω~(k2) for the length of
strong 3-query LDCs. In contrast to previous work, our techniques are purely
combinatorial and do not rely on a direct reduction to 2-query LDCs, opening
up a potentially different approach to analyzing 3-query LDCs.Comment: 10 page