Codes with Large Minimum Distance in Product Codes: Explicit Constructions and Bounds

2026-04-16

AI Summary

This study addresses the construction of explicit subcodes of Reed–Solomon product codes with enhanced minimum distance to improve error-correction capability. By integrating algebraic coding theory, product code structures, and combinatorial design techniques, the authors achieve the largest known minimum distances for subcodes of dimensions \( r^2 - 1 \), \( r^2 - 2 \), and at most \( 2r - 1 \), thereby establishing new upper bounds on their minimum distances. The work significantly reduces the required field size compared to prior constructions, attaining optimal or near-optimal distance performance for given dimensions and yielding an improved trade-off between dimension and minimum distance.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Feature Construction/Reformulation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Products of MDS codes are of major practical importance; for a recent example, they are used in Data Availability Sampling (DAS) in blockchain networks such as Celestia and as part of the Ethereum roadmap. This motivates us to consider subcodes of such codes with the goal of obtaining a larger minimum distance. In this paper, we present explicit constructions of subcodes of Reed--Solomon product codes, along with bounds on their minimum distance. In particular, they achieve an optimal or near-optimal dimension--distance tradeoff. For component codes of dimension $r$, our construction requires a field whose size is bounded linearly by the overall product code length, and attains the maximum possible minimum distance for subcode dimensions $r^2-1$, $r^2-2$, and all dimensions at most $2r-1$. Furthermore, we establish a new upper bound on the minimum distance of subcodes of the product of two codes with identical parameters.

Problem

• product codes
• minimum distance
• subcodes
• MDS codes
• dimension--distance tradeoff

Innovation

• product codes
• Reed–Solomon codes
• minimum distance
• explicit constructions
• subcodes