Explicit Constant-Alphabet Subspace Design Codes

TL;DR

This paper constructs explicit subspace design codes over constant-sized alphabets using expander-based methods, answering an open question and improving list-recovery parameters.

Contribution

It provides the first explicit constant-alphabet subspace design codes, generalizing prior work and leveraging the AEL framework for better parameters.

Findings

Constructed explicit constant-alphabet subspace design codes.

Codes share local properties of random linear codes.

Improved parameters for list-recovery.

Abstract

The subspace design property for additive codes is a higher-dimensional generalization of the minimum distance property. As shown recently by Brakensiek, Chen, Dhar and Zhang, it implies that the code has similar performance as random linear codes with respect to all "local properties". Explicit algebraic codes, such as folded Reed-Solomon and multiplicity codes, are known to have the subspace design property, but they need alphabet sizes that grow as a large polynomial in the block length. Constructing explicit constant-alphabet subspace design codes was subsequently posed as an open question in Brakensiek, Chen, Dhar and Zhang. In this work, we answer their question and give explicit constructions of subspace design codes over constant-sized alphabets, using the expander-based Alon-Edmonds-Luby (AEL) framework. This generalizes the recent work of Jeronimo and Shagrithaya, which showed that such codes share local properties of random linear codes. Our work obtains this consequence in a unified manner via the subspace design property. In addition, our approach yields some improvements in parameters for list-recovery.