Algebraic-Geometric Parvaresh--Vardy Subspace Designs and Rank Condensers

Abstract

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.