Near-Tight Bounds for 3-Query Locally Correctable Binary Linear Codes via Rainbow Cycles

TL;DR

This paper resolves the dimension bounds for binary linear 3-query LCCs by proving that for an $[n,k]$ binary linear $(3,\delta)$-LCC, $k \leqslant O(\delta^{-2} \log^2 n \cdot \log \log n)$, which is near-optimal given the $k = \Theta(\log^2 n)$ of degree-2 Reed-Muller codes. The authors achieve this through a direct analysis of the dual code's covering radius, converting the problem into sparse representations of generator rows via a compression/shifting mechanism that is facilitated by rainbow cycles in properly edge-colored graphs. They connect 3-LCC lower bounds to rainbow LDC lower bounds, deriving higher-query LCC bounds for odd $r \ge 5$ and clarifying the relationship between LCC and LDC lower bounds. The approach leverages recent rainbow-cycle breakthroughs (ABSZZ23) to yield modular, scalable arguments, offering a pathway to tighter bounds as rainbow-cycle results improve.

Abstract

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $δ> 0$ of adversarial errors must have dimension at most $O_δ(\log^2 n \cdot \log \log n)$. This is almost tight in view of quadratic Reed-Muller codes being a $3$-query locally correctable code (LCC) with dimension $Θ(\log^2 n)$. Our result improves, for the binary field case, the $O_δ(\log^8 n)$ bound obtained in the recent breakthrough of (Kothari and Manohar, 2023) (arXiv:2311.00558) (and the more recent improvement to $O_δ(\log^4 n)$ for binary linear codes announced in (Yankovitz, 2024)). Previous bounds for $3$-query linear LCCs proceed by constructing a $2$-query locally decodable code (LDC) from the $3$-query linear LCC/LDC and applying the strong bounds known for the former. Our approach is more direct and proceeds by bounding the covering radius of the dual code, borrowing inspiration from (Iceland and Samorodnitsky, 2018) (arXiv:1802.01184). That is, we show that if $x \mapsto (v_1 \cdot x, v_2 \cdot x, \ldots, v_n \cdot x)$ is an arbitrary encoding map $\mathbb{F}_2^k \to \mathbb{F}_2^n$ for the $3$-query LCC, then all vectors in $\mathbb{F}_2^k$ can be written as a $\widetilde{O}_δ(\log n)$-sparse linear combination of the $v_i$'s, which immediately implies $k \le \widetilde{O}_δ((\log n)^2)$. The proof of this fact proceeds by iteratively reducing the size of any arbitrary linear combination of at least $\widetildeΩ_δ(\log n)$ of the $v_i$'s. We achieve this using the recent breakthrough result of (Alon, Bucić, Sauermann, Zakharov, and Zamir, 2023) (arXiv:2309.04460) on the existence of rainbow cycles in properly edge-colored graphs, applied to graphs capturing the linear dependencies underlying the local correction property.

Figures (1)

• Figure 1: This figure indicates the cancellations that occur in our proof of \ref{['thm:intro-cov-rad']} via iterative refinement of the representation of an arbitrary vector $x \in \mathbb{F}_2^k$ as a sum more than $p(B+1) = \Omega(\log{n}\log\log{n})$ of the $v_i$'s. The nodes represent indices in $[n]$, with the gray nodes indicating 'canceled' nodes in the sum $\sum_{i \in I}{v_i}$, while the black nodes represent the 'active' nodes in the sum. The inner gray nodes in the pentagon and the square are cancellations resulting from \ref{['lem:shift-representation']}. The cancellation of the one outer gray node in common is the result of picking a common node between two 'shifts' $T_{\ell_1}'$ and $T_{\ell_2}'$ of the sets $T_{\ell_1}$ and $T_{\ell_2}$, which is key idea in the proof of \ref{['thm:intro-cov-rad']} from \ref{['lem:shift-representation']}. In the figure, a sum of $9$ terms (the indices $t_s$ corresponding to each of the $9$ colors) is compressed into a sum of $7$ terms (the black nodes).

Theorems & Definitions (25)

• Theorem 1.1: Main
• Theorem 1.2
• Theorem 1.3
• Definition 2.1: Binary Linear LCC, local decoder definition
• Definition 2.2: Binary Linear LCC, combinatorial definition
• Remark 2.1
• Definition 2.3: Rainbow Cycle
• Theorem 2.1: ABSZZ23, Theorem 1.1
• Theorem 3.1
• proof : Proof of \ref{['thm:main']} from \ref{['thm:compression']}