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.

翻译：局部可译码（LDC）是一种将$Σ^n$中的\emph{消息}编码为$Σ^m$中的\emph{码字}的错误校正码$C\colonΣ^n\rightarrow Σ^m$，其具备超快速解码算法。作为理论计算机科学多个领域的重要数学对象，目前对于具有恒定查询复杂度与恒定字母表规模的此类码，最优构造的码字长度$m$相对于$n$仍呈超多项式增长。Ben-Sasson、Goldreich、Harsha、Sudan与Vadhan（SICOMP 2006）在一项令人惊异的成果中，提出了在二进制字母表上构造具有恒定查询复杂度与几乎线性码字长度的松弛局部可译码（RLDC）的方法，并借此实现了概率可检测证明的显著改进构造。本研究针对标准汉明误差设定下的RLDC展开。我们证明了在二进制字母表上，即使允许自适应查询，进行$2$次查询的汉明RLDC的码长存在指数下界。该结果明确回答了Gur与Lachish（SICOMP 2021）提出的问题，并首次为RLDC建立了指数下界。结合Ben-Sasson等人的成果，我们的研究揭示了在某些恒定查询复杂度下码字长度呈现的“相变”型行为。这些下界的证明通过将RLDC转化为标准汉明LDC实现，其中对固定码字位的消息比特限制进行了精细分析。