Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
Abstract
Locally recoverable codes (LRCs) play an important role in distributed and cloud storage systems. The celebrated Tamo--Barg construction of optimal LRCs is based on the notion of \emph{good polynomials}. In this paper, we introduce the notion of \emph{good rational functions} as a natural generalization of good polynomials and develop a more general framework for constructing optimal LRCs. Since the attainable code lengths depend on the number of fibers of maximal size, we study this quantity using algebraic function field theory and Galois theory, identifying it with the number of totally split rational places in the associated function field extension. By analyzing the Galois closure, we establish an exact counting formula and derive sharp upper bounds for the number of ramified places and the genus, leading to explicit lower bounds for the number of totally split rational places. When the extension is Galois, we further characterize the number of totally split rational places in terms of the action of the Galois group on the projective line. Explicit constructions of good rational functions in both Galois and non-Galois settings are also presented. These rational functions yield optimal LRCs with lengths exceeding those attainable from good polynomials of the same degree, demonstrating the advantage of our framework over the classical Tamo--Barg construction.
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BibTeX
@misc{liu2026beyond,
title = {{Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions}},
author = {Hengfeng Liu and Sihem Mesnager and Chunming Tang and Xuemin Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.11465},
eprint = {2605.11465},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11465},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions
AU - Hengfeng Liu
AU - Sihem Mesnager
AU - Chunming Tang
AU - Xuemin Zheng
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2605.11465
UR - https://arxiv.org/abs/2605.11465
AB - Locally recoverable codes (LRCs) play an important role in distributed and cloud storage systems. The celebrated Tamo--Barg construction of optimal LRCs is based on the notion of \emph{good polynomials}. In this paper, we introduce the notion of \emph{good rational functions} as a natural generalization of good polynomials and develop a more general framework for constructing optimal LRCs. Since the attainable code lengths depend on the number of fibers of maximal size, we study this quantity using algebraic function field theory and Galois theory, identifying it with the number of totally split rational places in the associated function field extension. By analyzing the Galois closure, we establish an exact counting formula and derive sharp upper bounds for the number of ramified places and the genus, leading to explicit lower bounds for the number of totally split rational places. When the extension is Galois, we further characterize the number of totally split rational places in terms of the action of the Galois group on the projective line. Explicit constructions of good rational functions in both Galois and non-Galois settings are also presented. These rational functions yield optimal LRCs with lengths exceeding those attainable from good polynomials of the same degree, demonstrating the advantage of our framework over the classical Tamo--Barg construction.
ER -