发表机构
Tel Aviv University; Ben Gurion University（特拉维夫大学; 本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于代数几何Parvaresh-Vardy码的显式强子空间设计，在常数域大小下达到最优k依赖，并由此构造任意有限域上的秩凝聚器与无损秩提取器，推广了现有结果。
AI 中文摘要
子空间设计是$\mathbb{F}_q^k$的子空间$H_1,\ldots,H_n$的集合，其性质是任何低维子空间$W$与该集合的交集“不会太多”。子空间设计以及线性代数伪随机性中的相关对象已找到广泛的应用，范围从列表译码到算法的去随机化。我们在每个有限域上构造了显式的强子空间设计。在极端情况下，每个$H_i$的余维数$t$等于$W$的维数，对于每个常数域大小，我们的构造达到$n=\Omega(k)$，并在常数因子内匹配概率交集界。所有先前的构造要求域大小随$t$（或$k$）增长。我们的子空间设计还蕴含了在任意有限域上秩凝聚器的新构造。这一结果是首个在保持恒定输出熵率和恒定域大小的同时，实现对$k$的最优依赖的结果。作为应用，我们为秩为$r$的线性源构造无损秩提取器，适用于所有$r<q$，其参数与Guo、Raj、Shangguan和Zhang（FOCS '26）的参数匹配，从而将他们的结果推广到素数域和更小的域大小。我们的构造基于Parvaresh-Vardy码的代数几何版本（Parvaresh-Vardy FOCS '05, Guruswami ECCC '05），扩展了Guruswami、Umans和Vadhan（JACM '09）的凝聚器框架。我们将我们的构造视为对GUV构造的线性代数分析——针对仿射源——并推广到代数曲线上的函数。更具体地说，受Ta-Shma和Umans（CCC 12'）的启发，我们开发了一个两级求值方案，其中我们首先在扩展域点上对曲线上的函数进行求值，然后对相应的仿射线性多项式进行求值，以获得基域上的输出。
英文摘要
A subspace design is a collection of subspaces $H_1,\ldots,H_n$ of $\mathbb{F}_q^k$ with the property that no low-dimensional subspace $W$ intersects the collection "too much". Subspace designs and related objects in linear-algebraic pseudorandomness have found a broad range of applications, ranging from list decoding, to derandomizing algorithms. We construct explicit strong subspace designs over every finite field. In the extremal case where the co-dimension $t$ of each $H_i$ is equal to the dimension of $W$, for every constant field size our construction attains $n=Ω(k)$ and matches the probabilistic intersection bound up to a constant factor. All previous constructions required the field size to grow with $t$ (or $k$). Our subspace designs also imply new construction of rank condensers over arbitrary finite fields. This result is the first to achieve an optimal dependence on $k$ while maintaining both a constant output entropy rate and a constant field size. As an application, we construct lossless rank extractors for linear sources of rank $r$, for all $r < q$, with parameters matching those of Guo, Raj, Shangguan and Zhang (FOCS '26), thereby generalizing their result to prime fields and smaller field sizes. Our construction is based on an algebraic-geometric version of the Parvaresh-Vardy codes (Parvaresh-Vardy FOCS '05, Guruswami ECCC '05), extending the framework underlying the condensers of Guruswami, Umans and Vadhan (JACM '09). We view our construction as a linear-algebraic analysis - tailored to affine sources - of the GUV construction, generalized to functions over algebraic curves. More specifically, inspired by Ta-Shma and Umans (CCC 12') we develop a two-level evaluation scheme, where we first evaluate a function on a curve at extension-field points, and then evaluate a corresponding affine-linear polynomial to obtain outputs over the base field.
