A linear-in-q range of dimensions for the MDS conjecture over 𝔽_q in odd characteristic

Abstract

Let q be a power of an odd prime p. We prove the MDS conjecture over 𝔽_q in every dimension k satisfying 2leqslant kleqslant B(p,q) or q+2−B(p,q)leqslant kleqslant q, B(p,q)=⌊((p−2)q+6p−10)/(2p−3)⌋. For fixed p, the first interval is linear in q; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic p>0. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at |G|=2k−3+n it proves Chowdhury's full-row-rank conjecture without the q-dependent restriction. Specialization to 𝔽_q, together with the Ball--Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power q, every normal rational curve in PG(N,q) is complete for 2leqslant Nleqslant q−2, and every projective Reed--Solomon code of length q+1 and dimension 2leqslant kleqslant q−2 has covering radius q−k.

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BibTeX RIS

BibTeX

@misc{fan2026linearinq,
  title = {{A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic}},
  author = {Xiang Fan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03740},
  eprint = {2610.03740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03740},
}

RIS

TY  - UNPB
TI  - A linear-in-q range of dimensions for the MDS conjecture over 𝔽_q in odd characteristic
AU  - Xiang Fan
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.03740
UR  - https://arxiv.org/abs/2610.03740
AB  - Let q be a power of an odd prime p. We prove the MDS conjecture over 𝔽_q in every dimension k satisfying 2leqslant kleqslant B(p,q) or q+2−B(p,q)leqslant kleqslant q, B(p,q)=⌊((p−2)q+6p−10)/(2p−3)⌋. For fixed p, the first interval is linear in q; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic p>0. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at |G|=2k−3+n it proves Chowdhury's full-row-rank conjecture without the q-dependent restriction. Specialization to 𝔽_q, together with the Ball--Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power q, every normal rational curve in PG(N,q) is complete for 2leqslant Nleqslant q−2, and every projective Reed--Solomon code of length q+1 and dimension 2leqslant kleqslant q−2 has covering radius q−k.
ER  -

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