小域上Reed-Muller码的列表译码
List decoding Reed-Muller codes over small fields
摘要：
一个代码的列表解码问题要求最大半径，直到该半径的任何球只包含一个常数数量的码字。列表解码半径甚至对于研究得很好的码（如Reed-Solomon码或Reed-Muller码）也没有很好的理解。修复有限域$\mathbb{F}$。Reed-Muller码$\mathrm{RM}_{\mathbb{F}}(n,d)$由$\mathbb{F}$上的$n$ -变量次数- $d$多项式定义。本文研究了常素域$\mathbb{F}=\mathbb{F}_p$、常度$d$和大$n$上Reed-Muller码的列表译码半径。我们证明了列表译码半径等于码的最小距离。也就是说，如果我们用$δ(d)$表示$\mathrm{RM}_{\mathbb{F}}(n,d)$的归一化最小距离，则半径为$δ(d)-\varepsilon$的任何球中的码字的数量由独立于$n$的$c=c(p,d,\varepsilon)$限定。这解决了Gopalan-Klivans-Zuckerman [STOC 2008]的一个猜想，他在其他结果中证明了$\mathbb{F}=\mathbb{F}_2$的特殊情况;并扩展了Gopalan [FOCS 2010]的工作，他在$d=2$的情况下证明了该猜想。我们还分析了球的半径超过最小距离的代码的码字的数量。对于$e \leq d$，我们证明了在半径为$δ(e) - \varepsilon$的球中$\mathrm{RM}_{\mathbb{F}}(n,d)$的码字数由$\exp(c \cdot n^{d-e})$限定，其中$c=c(p,d,\varepsilon)$与$n$无关。对$n$的依赖性很强。这扩展了Kaufman-Lovett-Porat [IEEE Inf. Theory 2012]的工作，他证明了$\mathbb{F}_2$上的类似界限。证明依赖于几个新的成分：弗里兹-Kannan弱正则性的一般函数空间的扩展，高阶傅立叶分析，以及Schwartz-Zippel引理的多项式组合的扩展。
英文摘要：
The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes. Fix a finite field $\mathbb{F}$. The Reed-Muller code $\mathrm{RM}_{\mathbb{F}}(n,d)$ is defined by $n$-variate degree-$d$ polynomials over $\mathbb{F}$. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field $\mathbb{F}=\mathbb{F}_p$, constant degree $d$ and large $n$. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by $δ(d)$ the normalized minimal distance of $\mathrm{RM}_{\mathbb{F}}(n,d)$, then the number of codewords in any ball of radius $δ(d)-\varepsilon$ is bounded by $c=c(p,d,\varepsilon)$ independent of $n$. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008], who among other results proved it in the special case of $\mathbb{F}=\mathbb{F}_2$; and extends the work of Gopalan [FOCS 2010] who proved the conjecture in the case of $d=2$. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$, we show that the number of codewords of $\mathrm{RM}_{\mathbb{F}}(n,d)$ in a ball of radius $δ(e) - \varepsilon$ is bounded by $\exp(c \cdot n^{d-e})$, where $c=c(p,d,\varepsilon)$ is independent of $n$. The dependence on $n$ is tight. This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who proved similar bounds over $\mathbb{F}_2$. The proof relies on several new ingredients: an extension of the Frieze-Kannan weak regularity to general function spaces, higher-order Fourier analysis, and an extension of the Schwartz-Zippel lemma to compositions of polynomials.
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