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LRCS: Duality, LP bounds, and field size.
Anina Gruica1, Benjamin Jany2, Alberto Ravagnani2
1Technical University of Denmark, Kongens Lyngby, Denmark.
Designs, Codes, and Cryptography
|April 28, 2026
Summary
This study introduces a duality theory for locally recoverable codes (LRCs), establishing new parameter bounds. It reveals that optimal LRCs require small minimum distance and block length relative to the field size.
Area of Science:
- Coding Theory
- Information Theory
- Discrete Mathematics
Background:
- Locally Recoverable Codes (LRCs) are essential for distributed storage systems, offering fault tolerance.
- Existing bounds on LRC parameters do not fully capture their locality properties.
- The MacWilliams identity is a fundamental tool in coding theory, relating code properties to their duals.
Purpose of the Study:
- To develop a duality theory for locally recoverable codes (LRCs).
- To establish new and improved bounds on the parameters of LRCs.
- To investigate the existence of optimal LRCs over small fields.
Main Methods:
- Development of a duality theory for LRCs.
- Introduction of a refined weight distribution capturing locality.
- Application of a MacWilliams-type identity for deriving LP-type bounds.
- Utilizing dual distance bounds and generalized weights.
Main Results:
- A new duality theory for LRCs is established.
- An LP-type bound is derived, outperforming existing bounds in certain cases.
- Non-existence results for optimal LRCs over small fields are obtained.
- It is shown that optimal LRCs necessitate small minimum distance and block length relative to field size.
Conclusions:
- The developed duality theory provides a powerful framework for analyzing LRCs.
- The new bounds offer tighter constraints on LRC parameters.
- The findings have implications for the design of efficient and robust error-correcting codes for storage systems.
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