eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-01-06
85:1
85:8
10.4230/LIPIcs.ITCS.2020.85
article
Combinatorial Lower Bounds for 3-Query LDCs
Bhattacharyya, Arnab
1
Chandran, L. Sunil
2
Ghoshal, Suprovat
2
National University of Singapore, Singapore
Indian Institute of Science, Bangalore, India
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 x_i 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 LDC’s of dimension k and length n, the best known bounds are: 2^{k^o(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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol151-itcs2020/LIPIcs.ITCS.2020.85/LIPIcs.ITCS.2020.85.pdf
Coding theory
Graph theory
Hypergraphs