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PreprintarXiv licencearXiv

On the maximum length of cyclic MDS codes

  • Gaojun Luo
  • Xiwang Cao
  • Shitao Li
  • Yang Li
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Abstract

We study the maximum length of nontrivial simple-root cyclic maximum distance separable (MDS) codes over 𝔽q. We prove n≀ q+1 if gcd(n,q2βˆ’1)>2, or if the dimension is even and gcd(n,q2βˆ’1)>1. The bound also holds for all lengths whose prime factors lie in {2,3,5,7}. We obtain further cases involving primes β„“β‰₯11 with (β„“βˆ’1)/2 prime, under conditions on the exponents and the characteristic. The proofs combine Fourier supports, locality bounds and field descent with arithmetic restrictions on multiplicative orders and prime-adic valuations.

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Cite

BibTeXRIS
BibTeX
@misc{luo2026maximum,
  title = {{On the maximum length of cyclic MDS codes}},
  author = {Gaojun Luo and Xiwang Cao and Shitao Li and Yang Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29810},
  eprint = {2610.29810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29810},
}
RIS
TY  - UNPB
TI  - On the maximum length of cyclic MDS codes
AU  - Gaojun Luo
AU  - Xiwang Cao
AU  - Shitao Li
AU  - Yang Li
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29810
UR  - https://arxiv.org/abs/2610.29810
AB  - We study the maximum length of nontrivial simple-root cyclic maximum distance separable (MDS) codes over 𝔽_q. We prove n≀ q+1 if gcd(n,q^2βˆ’1)>2, or if the dimension is even and gcd(n,q^2βˆ’1)>1. The bound also holds for all lengths whose prime factors lie in {2,3,5,7}. We obtain further cases involving primes β„“β‰₯11 with (β„“βˆ’1)/2 prime, under conditions on the exponents and the characteristic. The proofs combine Fourier supports, locality bounds and field descent with arithmetic restrictions on multiplicative orders and prime-adic valuations.
ER  -