Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912934170460160 |
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| author | Block, Alexander R. Blocki, Jeremiah Cheng, Kuan Grigorescu, Elena Li, Xin Zheng, Yu Zhu, Minshen |
| author_facet | Block, Alexander R. Blocki, Jeremiah Cheng, Kuan Grigorescu, Elena Li, Xin Zheng, Yu Zhu, Minshen |
| contents | Locally Decodable Codes (LDCs) are error-correcting codes $C\colonΣ^n\rightarrow Σ^m,$ encoding \emph{messages} in $Σ^n$ to \emph{codewords} in $Σ^m$, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size.
In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs.
In this work, we study RLDCs in the standard Hamming-error setting. We prove an exponential lower bound on the length of Hamming RLDCs making $2$ queries (even adaptively) over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a ``phase-transition''-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20278 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes Block, Alexander R. Blocki, Jeremiah Cheng, Kuan Grigorescu, Elena Li, Xin Zheng, Yu Zhu, Minshen Information Theory Computational Complexity Locally Decodable Codes (LDCs) are error-correcting codes $C\colonΣ^n\rightarrow Σ^m,$ encoding \emph{messages} in $Σ^n$ to \emph{codewords} in $Σ^m$, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting. We prove an exponential lower bound on the length of Hamming RLDCs making $2$ queries (even adaptively) over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a ``phase-transition''-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits. |
| title | Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes |
| topic | Information Theory Computational Complexity |
| url | https://arxiv.org/abs/2602.20278 |