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：一组集合对于某个性质满足$(δ,\varepsilon)$-邻近距离，若对于集合中任意子集，要么(i)该子集中所有成员在（相对）汉明距离上均$δ$-接近该性质，要么(ii)仅有至多$\varepsilon$比例的成员$δ$-接近该性质。在一项开创性工作中，Ben-Sasson等人证明了仿射子空间集合对于Reed–Solomon（RS）码字性质具有$(δ,\varepsilon)$-邻近距离，其中$δ$可达列表解码的所谓Johnson界。其技术依赖于RS码的Guruswami–Sudan列表解码算法，该算法保证在Johnson界范围内有效。折叠Reed–Solomon（FRS）码已知能实现最优列表解码半径$δ$（称为容量区域），且已发展了丰富的FRS码列表解码算法。自然的问题在于：FRS码能否展现类似的$(δ,\varepsilon)$-邻近距离，且可达到所谓的最优容量区域？本文给出肯定回答（该框架可自然推广至合适的子空间设计码）。理解FRS码邻近距离的另一动因源于近期结果： [BCDZ'25]表明FRS码具有类似随机线性码的性质，而[LMS'25]此前已证明随机线性码与随机评估点RS码的性质相关，此外还有基于AEL的常数大小字母表码 [JS'25]。