# Subspace Designs based on Algebraic Function Fields

## TL;DR

This paper introduces a new method for constructing subspace designs over any field using algebraic function fields, extending previous polynomial-based approaches and enabling applications like dimension expanders over all fields.

## Contribution

It extends subspace design constructions to arbitrary fields by leveraging algebraic function fields, broadening their applicability and improving their utility in linear-algebraic pseudorandomness.

## Key findings

- Constructed subspace designs over any field with slightly weaker parameters.
- Applied the new designs to produce dimension expanders over all fields.
- Achieved logarithmic degree and expansion for large subspaces.

## Abstract

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically $q \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over $\F^n$ for any field $\F$, with logarithmic degree and expansion guarantee for subspaces of dimension $\Omega(n/(\log \log n))$.