Improving 3-query RLDCs or proving matching lower bounds

Determine whether 3-query relaxed locally decodable codes with constant decoding radius, constant distance, and constant alphabet can achieve sub-quadratic blocklength in the message length k. Either construct such sub-quadratic 3-query RLDCs or prove lower bounds that match the nearly quadratic length achieved by the construction in this paper.

Background

The paper presents 3-query RLDCs with constant alphabet, constant decoding radius, and constant distance whose blocklength is nearly quadratic in the message length, yielding a separation from 3-query LDCs.

The authors explicitly state that it is unknown whether better 3-query RLDCs exist and call for either stronger lower bounds matching their performance or constructions achieving sub-quadratic length.

References

Lastly, our codes from~\Cref{thm:RLDC_intro} have $3$ queries and nearly quadratic length, and in a companion paper we show how to generalize these codes for any odd number of queries $q 3$. We do not know if better $3$-query RLDCs exist, and it would be interesting to either prove lower bounds that match the performance of our codes, or come up with sub-quadratic $3$-query RLDCs.

— 3-Query RLDCs are Strictly Stronger than 3-Query LDCs (2512.12960 - Gur et al., 15 Dec 2025) in Discussion and Open Problems