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Gabidulin Codes Achieve List Decoding Capacity with an Order-Optimal Column-To-Row Ratio

Authors Zeyu Guo , Chaoping Xing , Chen Yuan , Zihan Zhang

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
Keywords
  • coding theory
  • error-correcting codes
  • Gabidulin codes
  • rank-metric codes

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Abstract

In this paper, we show that random Gabidulin codes of block length n and rate R achieve the (average-radius) list decoding capacity of radius 1-R-ε in the rank metric with an order-optimal column-to-row ratio of O(ε). This extends the recent work of Guo, Xing, Yuan, and Zhang (FOCS 2024), improving their column-to-row ratio from O(ε/n) to O(ε). For completeness, we also establish a matching lower bound on the column-to-row ratio for capacity-achieving Gabidulin codes in the rank metric.
Our proof techniques build on the work of Guo and Zhang (FOCS 2023), who showed that randomly punctured Reed-Solomon codes over fields of quadratic size attain the generalized Singleton bound of Shangguan and Tamo (STOC 2020) in the Hamming metric. The proof of our lower bound follows the method of Alrabiah, Guruswami, and Li (SODA 2024) for codes in the Hamming metric.

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Zeyu Guo, Chaoping Xing, Chen Yuan, and Zihan Zhang. Gabidulin Codes Achieve List Decoding Capacity with an Order-Optimal Column-To-Row Ratio. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 43:1-43:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.43

Author Details

Zeyu Guo
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Chaoping Xing
  • School of Electronic Information and Electric Engineering, Shanghai Jiao Tong University, China
Chen Yuan
  • School of Electronic Information and Electric Engineering, Shanghai Jiao Tong University, China
Zihan Zhang
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA

Funding

  • Guo, Zeyu: Supported by NSF grant CCF-2440926.
  • Zhang, Zihan: Supported by NSF grant CCF-2440926.

Acknowledgements

The authors thank the anonymous RANDOM 2025 reviewers for their helpful comments.

References

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