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.
@InProceedings{bhattacharyya_et_al:LIPIcs.ITCS.2020.85, author = {Bhattacharyya, Arnab and Chandran, L. Sunil and Ghoshal, Suprovat}, title = {{Combinatorial Lower Bounds for 3-Query LDCs}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {85:1--85:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.85}, URN = {urn:nbn:de:0030-drops-117704}, doi = {10.4230/LIPIcs.ITCS.2020.85}, annote = {Keywords: Coding theory, Graph theory, Hypergraphs} }
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