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PreprintCC BY 4.0arXiv

Length Bounds for MDS Algebraic Geometry Codes of Arbitrary Positive Genus

  • Haojie Chen
  • Chuangqiang Hu
  • Guorui Ma
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Abstract

We study the length of maximum distance separable algebraic geometry codes from curves of arbitrary positive genus. For fixed genus, we prove that the asymptotic ratio of the length to the field size is at most 1/2 when both the dimension and the codimension grow faster than the logarithm of the field size. We also give explicit bounds under logarithmic dimension conditions. The bounds hold in every characteristic and are uniform in the curve and the evaluation set. The proof combines character sums on the Jacobian with an estimate for sums of distinct points.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026length,
  title = {{Length Bounds for MDS Algebraic Geometry Codes of Arbitrary Positive Genus}},
  author = {Haojie Chen and Chuangqiang Hu and Guorui Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39434},
  eprint = {2610.39434},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39434},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Length Bounds for MDS Algebraic Geometry Codes of Arbitrary Positive Genus
AU  - Haojie Chen
AU  - Chuangqiang Hu
AU  - Guorui Ma
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39434
UR  - https://arxiv.org/abs/2610.39434
AB  - We study the length of maximum distance separable algebraic geometry codes from curves of arbitrary positive genus. For fixed genus, we prove that the asymptotic ratio of the length to the field size is at most 1/2 when both the dimension and the codimension grow faster than the logarithm of the field size. We also give explicit bounds under logarithmic dimension conditions. The bounds hold in every characteristic and are uniform in the curve and the evaluation set. The proof combines character sums on the Jacobian with an estimate for sums of distinct points.
ER  -