A collection of sets satisfies a $(δ,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $δ$-close to the property in (relative) Hamming distance, or (ii) only a small $\varepsilon$-fraction of members are $δ$-close to the property. In a seminal work, Ben-Sasson \textit{et al.}\ showed that the collection of affine subspaces exhibits a $(δ,\varepsilon)$-proximity gap with respect to the property of being Reed--Solomon (RS) codewords with $δ$ up to the so-called Johnson bound for list decoding. Their technique relies on the Guruswami--Sudan list decoding algorithm for RS codes, which is guaranteed to work in the Johnson bound regime. Folded Reed--Solomon (FRS) codes are known to achieve the optimal list decoding radius $δ$, a regime known as capacity. Moreover, a rich line of list decoding algorithms was developed for FRS codes. It is then natural to ask if FRS codes can be shown to exhibit an analogous $(δ,\varepsilon)$-proximity gap, but up to the so-called optimal capacity regime. We answer this question in the affirmative (and the framework naturally applies more generally to suitable subspace-design codes). An additional motivation to understand proximity gaps for FRS codes is the recent results [BCDZ'25] showing that they exhibit properties similar to random linear codes, which were previously shown to be related to properties of RS codes with random evaluation points in [LMS'25], as well as codes over constant-size alphabet based on AEL [JS'25].

翻译：若一个集合族关于某个性质满足$(δ,\varepsilon)$-邻近间隙，则对于该族中的任意集合，要么（i）集合中所有成员在（相对）汉明距离下均$δ$-接近该性质，要么（ii）仅有少量$\varepsilon$比例的成员$δ$-接近该性质。在开创性工作中，Ben-Sasson等人证明了仿射子空间族关于里德-所罗门码字性质具有$(δ,\varepsilon)$-邻近间隙，其中$δ$可达列表解码的Johnson界。其证明技术依赖于Guruswami-Sudan里德-所罗门码列表解码算法，该算法在Johnson界范围内具有理论保证。已知折叠里德-所罗门码能达到最优列表解码半径$δ$，即达到容量界。此外，针对折叠里德-所罗门码已发展出一系列丰富的列表解码算法。这自然引出一个问题：折叠里德-所罗门码是否也能展现出类似的$(δ,\varepsilon)$-邻近间隙，且能达到最优容量界？我们对此问题给出了肯定回答（该框架自然可推广至适用于更一般的子空间设计码）。研究折叠里德-所罗门码邻近间隙的另一动机来自近期成果[BCDZ'25]，该研究表明折叠里德-所罗门码展现出与随机线性码相似的性质，而此前研究[LMS'25]已证明这类性质与随机评估点里德-所罗门码相关，[JS'25]也基于AEL构建了具有类似性质的常字母表码。