Relaxed vs. Full Local Decodability with Few Queries: Equivalence and Separations for Linear Codes

A locally decodable code (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ is an error-correcting code that allows one to recover any bit of the original message with good probability while only reading a small number of bits from a corrupted codeword. A relaxed locally decodable code (RLDC) is a weaker notion where the decoder is additionally allowed to abort and output a special symbol $\bot$ if it detects an error. For a large constant number of queries $q$, there is a large gap between the blocklength $n$ of the best $q$-query LDC and the best $q$-query RLDC. Existing constructions of RLDCs achieve polynomial length $n = k^{1 + O(1/q)}$, while the best-known $q$-LDCs only achieve subexponential length $n = 2^{k^{o(1)}}$. On the other hand, for $q = 2$, it is known that RLDCs and LDCs are equivalent. We thus ask the question: what is the smallest $q$ such that there exists a $q$-RLDC that is not a $q$-LDC? In this work, we show that any linear $3$-query RLDC is in fact a $3$-LDC, i.e., linear RLDCs and LDCs are equivalent at $3$ queries. More generally, we show for any constant $q$, there is a soundness error threshold $s(q)$ such that any linear $q$-RLDC with soundness error below this threshold must be a $q$-LDC. This implies that linear RLDCs cannot have "strong soundness" -- a stricter condition satisfied by linear LDCs that says the soundness error is proportional to the fraction of errors in the corrupted codeword -- unless they are simply LDCs. In addition, we give simple constructions of linear $15$-query RLDCs that are not $q$-LDCs for any constant $q$, showing that for $q = 15$, linear RLDCs and LDCs are not equivalent. We also prove nearly identical results for locally correctable codes and their corresponding relaxed counterpart.