Technology Category
Search and Optimization: Mixed Discrete/Continuous SearchMachine Learning: Probabilistic Circuits and Graphical ModelsConstraint Satisfaction and Optimization: Distributed CSP/Optimization
Explicit Constant-Alphabet Subspace Design Codes
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
subspace design
constant-alphabet codes
explicit constructions
additive codes
list-recovery
Innovation
Methods, ideas, or system contributions that make the work stand out.
subspace design
constant-alphabet codes
explicit construction
AEL framework
list-recovery
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