Combinatorial Lower Bounds for 3-Query LDCs
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.
Coding theory
Graph theory
Hypergraphs
Theory of computation~Error-correcting codes
85:1-85:8
Regular Paper
AB thanks Sivakanth Gopi, Nikhil Srivastava, and Luca Trevisan for many useful discussions about this problem.
Arnab
Bhattacharyya
Arnab Bhattacharyya
National University of Singapore, Singapore
Partially supported by NUS Startup Grant R-252-000-A33-133.
L. Sunil
Chandran
L. Sunil Chandran
Indian Institute of Science, Bangalore, India
Suprovat
Ghoshal
Suprovat Ghoshal
Indian Institute of Science, Bangalore, India
10.4230/LIPIcs.ITCS.2020.85
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