← 研踪 PaperTrailarXiv:2611.03145 · v1
Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs
首次提交 2026-11-02 21:53
版本更新 2026-11-02 21:53
雷达观测 2026-11-04 23:41
摘要
We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{Ω(\sqrt{k})}$. This improves on the prior best lower bound of $2^{Ω(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed-Muller codes, which achieve length $n = 2^{O(\sqrt{k})}$, is optimal up to constant factors in the exponent. Our proof builds on the approach of [KM24a], further refined in [Yan24,KM24b], that uses "chain derived equations" in combination with Kikuchi graphs. The crux of our argument is to show that a certain edge-coloured Kikuchi graph has a unique colour cycle: a cycle in which some colour is used exactly once. We do this by controlling the distribution of colours at each vertex in the graph using a random walk mixing argument and a novel hypergraph decomposition based on edge expansion of the hypergraph's "flattening" into a graph.
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v1 · 2026-11-02 21:53 · 最新观测版本
Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs
We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{Ω(\sqrt{k})}$. This improves on the prior best lower bound of $2^{Ω(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed-Muller codes, which achieve length $n = 2^{O(\sqrt{k})}$, is optimal up to constant factors in the exponent. Our proof builds on the approach of [KM24a], further refined in [Yan24,KM24b], that uses "chain derived equations" in combination with Kikuchi graphs. The crux of our argument is to show that a certain edge-coloured Kikuchi graph has a unique colour cycle: a cycle in which some colour is used exactly once. We do this by controlling the distribution of colours at each vertex in the graph using a random walk mixing argument and a novel hypergraph decomposition based on edge expansion of the hypergraph's "flattening" into a graph.
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