We prove the existence of Reed-Solomon codes of any desired rate $R \in (0,1)$ that are combinatorially list-decodable up to a radius approaching $1-R$, which is the information-theoretic limit. This is established by starting with the full-length $[q,k]_q$ Reed-Solomon code over a field $\mathbb F_q$ that is polynomially larger than the desired dimension $k$, and "puncturing" it by including $k/R$ randomly chosen codeword positions. Our puncturing result is more general and applies to any code with large minimum distance: we show that a random rate $R$ puncturing of an $\mathbb F_q$-linear "mother" code whose relative distance is close enough to $1-1/q$ is list-decodable up to a radius approaching the $q$-ary list-decoding capacity bound $h_q^{-1}(1-R)$. In fact, for large $q$, or under a stronger assumption of low-bias of the mother-code, we prove that the threshold rate for list-decodability with a specific list-size (and more generally, any "local" property) of the random puncturing approaches that of fully random linear codes. Thus, all current (and future) list-decodability bounds shown for random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) code. This can be viewed as a general derandomization result applicable to random linear codes. To obtain our conclusion about Reed-Solomon codes, we establish some hashing properties of field trace maps that allow us to reduce the list-decodability of RS codes to its associated trace (dual-BCH) code, and then apply our puncturing theorem to the latter. Our approach implies, essentially for free, optimal rate list-recoverability of punctured RS codes as well.

中文翻译：

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打孔大距离码，和许多里德-所罗门码，实现列表解码能力

我们证明了任何所需速率 $R \in (0,1)$ 的 Reed-Solomon 码的存在，这些码可以组合列表解码，半径接近 $1-R$，这是信息论的极限。这是通过从多项式大于所需维度 $k$ 的字段 $\mathbb F_q$ 上的全长 $[q,k]_q$ Reed-Solomon 代码开始建立的，并通过包含 $ k/R$ 随机选择的码字位置。我们的打孔结果更一般，适用于任何具有大的最小距离的代码：我们表明，相对距离足够接近 $1-1/ 的 $\mathbb F_q$-线性“母”代码的随机速率 $R$ q$ 是列表可解码的，半径接近 $q$-ary 列表解码容量界限 $h_q^{-1}(1-R)$。事实上，对于大的 $q$，或者在母码低偏差的更强假设下，我们证明具有特定列表大小（更一般地，任何“本地”属性）的随机穿孔的列表可解码性阈值率接近完全随机线性代码。因此，为随机线性代码显示的所有当前（和未来）列表可解码性边界自动扩展到任何低偏差（或大字母表）代码的随机穿孔。这可以看作是适用于随机线性码的一般去随机化结果。为了获得关于 Reed-Solomon 码的结论，我们建立了一些场迹映射的散列属性，这些属性允许我们将 RS 码的列表可解码性减少到其相关的迹（双 BCH）码，然后将我们的穿孔定理应用于后者。我们的方法意味着，基本上是免费的，