Explicit Constant-Alphabet Subspace Design Codes

2026-04-16

AI Summary

This work resolves an open problem posed by Brakensiek et al. by presenting the first explicit construction of error-correcting codes with subspace design properties over a constant-sized alphabet, whereas prior constructions required alphabets growing polynomially with the code length. Building upon the Alon–Edmonds–Luby (AEL) expander framework, the proposed construction seamlessly integrates algebraic coding techniques with combinatorial expander graphs. The resulting codes retain local properties comparable to those of random linear codes while achieving significantly improved list-recovery parameters.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• subspace design
• constant-alphabet codes
• explicit constructions
• additive codes
• list-recovery

Innovation

• subspace design
• constant-alphabet codes
• explicit construction
• AEL framework
• list-recovery