容量处随机线性码的列表大小
The list size of random linear codes at capacity
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中文总结 AI 辅助
本文确定了容量处随机线性码列表大小的精确值，通过新的熵不等式和向日葵引理证明了阈值速率的两侧界，解决了所有素数幂情形下的开放问题。
中文摘要 AI 辅助
设 $C \le \mathbb{F}_q^n$ 为速率 $1 - h_q(\rho) - \varepsilon$ 的一致随机 $\mathbb{F}_q$-线性码，并设 $L^*(C,\rho)$ 为使得相对半径 $\rho$ 的每个 Hamming 球至多包含 $C$ 的 $L$ 个码字的最小 $L$。自 Guruswami、Håstad 和 Kopparty 以及 Guruswami 和 Narayanan 的工作以来，已知 $L^* = \Theta_{q,\rho}(1/\varepsilon)$。Guruswami、Li、Mosheiff、Resch、Silas 和 Wootters 证明了对于所有 $q$，$1/\varepsilon$ 前面的常数至少为 $h_q(\rho)$，并给出了一个仅适用于 $q = 2$ 的上界，该上界将 $L^*$ 限定在该情形下的三个连续整数之内。但对于 $q \ge 3$，此前未知具有正确常数的上界。我们确定了每个素数幂 $q$ 的 $L^*$。令 $\zeta := h_q(\rho)/\varepsilon$。对于每个足够小的 $\varepsilon$，以概率 $1-o(1)$（对 $C$ 的选择而言），$$L^*(C,\rho) = \lceil \zeta \rceil,$$ 除非 $\zeta$ 的小数部分至多为 $q^{-\Omega_{q,\rho}(\zeta)}$，在这种情况下 $L^*(C,\rho)$ 为 $\lfloor \zeta \rfloor$ 或 $\lfloor \zeta \rfloor + 1$。根据 Mosheiff、Resch、Ron-Zewi、Silas 和 Wootters 对随机线性码的阈值刻画，两个界都归结为对单一量 $V(q,L,\rho)$ 的双侧估计，其中 $1-V(q,L,\rho)$ 是 $(\rho,L)$-列表可解码性的阈值速率。我们证明对所有大的 $L$：$$h_q(\rho)(1 + 1/L) - q^{-\Omega_{q,\rho}(L)} \le V(q,L,\rho) \le h_q(\rho)(1 + 1/L)。$$ 上界依赖于一个新的熵不等式，该不等式针对在成对非成比例的线性约束下的稀疏随机向量，并借助 Erdős-Rado 向日葵引理证明。下界是对 Guruswami、Li、Mosheiff、Resch、Silas 和 Wootters 引入的分布的精确分析。
英文摘要
Let $C \le \mathbb{F}_q^n$ be a uniformly random $\mathbb{F}_q$-linear code of rate $1 - h_q(ρ) - \varepsilon$, and let $L^*(C,ρ)$ be the least $L$ such that every Hamming ball of relative radius $ρ$ contains at most $L$ codewords of $C$. That $L^* = Θ_{q,ρ}(1/\varepsilon)$ has been known since work of Guruswami, Håstad and Kopparty and of Guruswami and Narayanan. Guruswami, Li, Mosheiff, Resch, Silas and Wootters proved that the constant in front of $1/\varepsilon$ is at least $h_q(ρ)$ for all $q$, along with an upper bound special to $q = 2$ which narrowed $L^*$ to within three consecutive integers in that case. But for $q \ge 3$ no upper bound with the correct constant was known. We determine $L^*$ for every prime power $q$. Let $ζ:= h_q(ρ)/\varepsilon$. For every sufficiently small $\varepsilon$, with probability $1-o(1)$ over the choice of $C$, $$L^*(C,ρ) = \lceil ζ\rceil,$$ unless the fractional part of $ζ$ is at most $q^{-Ω_{q,ρ}(ζ)}$, in which case $L^*(C,ρ)$ is $\lfloor ζ\rfloor$ or $\lfloor ζ\rfloor + 1$. By the threshold characterization of random linear codes due to Mosheiff, Resch, Ron-Zewi, Silas and Wootters, both bounds reduce to a two-sided estimate of a single quantity $V(q,L,ρ)$, where $1-V(q,L,ρ)$ is the threshold rate for $(ρ,L)$-list-decodability. We prove for all large $L$: $$h_q(ρ)(1 + 1/L) - q^{-Ω_{q,ρ}(L)} \le V(q,L,ρ) \le h_q(ρ)(1 + 1/L).$$ The upper bound rests on a new entropy inequality for sparse random vectors under pairwise non-proportional linear constraints, proved with the Erdős-Rado sunflower lemma. The lower bound is an exact analysis of the distribution introduced by Guruswami, Li, Mosheiff, Resch, Silas and Wootters.
