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

2026-09-26

AI Summary

This study addresses the challenge of explicitly constructing strong subspace designs and rank condensers over finite fields. By leveraging algebraic-geometric Parvaresh–Vardy codes combined with a two-level evaluation scheme and linear-algebraic pseudorandomness techniques, this work proposes a novel method for building optimal subspace designs and rank condensers over arbitrary finite fields. For the first time, it achieves optimal parameters matching the probabilistic bound under constant field sizes, with results further extended to prime fields. The resulting rank condensers exhibit an optimal dependence on the dimension parameter k. Moreover, this approach yields lossless rank extractors, thereby overcoming prior theoretical limitations in the field.

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 $n=\Omega(k)$ 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.

Problem

• subspace design
• rank condenser
• finite field
• rank extractor
• linear-algebraic pseudorandomness

Innovation

• Subspace Designs
• Rank Condensers
• Algebraic-Geometric Parvaresh-Vardy Codes
• Two-Level Evaluation Scheme
• Lossless Rank Extractors