On Almost MDS codes holding (n−k−1)-designs and an open problem on Witt design

Abstract

Let 𝒞 be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an (n−k−1)-$(n,n-k,λ)$ design. We first determine the parameters of the dual code 𝒞^⊥. We also characterize the design and its complementary design supported by the minimum weight codewords of 𝒞^⊥. In particular, we present explicit formulas for the weight enumerators of all subset codes of 𝒞^⊥ with length at least k+1. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter λ, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.

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BibTeX RIS

BibTeX

@misc{wei2026almost,
  title = {{On Almost MDS codes holding $(n-k-1)$-designs and an open problem on Witt design}},
  author = {Huakai Wei and Shiyuan Qiang and Shaofang Hong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29744},
  eprint = {2610.29744},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29744},
}

RIS

TY  - UNPB
TI  - On Almost MDS codes holding (n−k−1)-designs and an open problem on Witt design
AU  - Huakai Wei
AU  - Shiyuan Qiang
AU  - Shaofang Hong
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29744
UR  - https://arxiv.org/abs/2610.29744
AB  - Let 𝒞 be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an (n−k−1)-$(n,n-k,λ)$ design. We first determine the parameters of the dual code 𝒞^⊥. We also characterize the design and its complementary design supported by the minimum weight codewords of 𝒞^⊥. In particular, we present explicit formulas for the weight enumerators of all subset codes of 𝒞^⊥ with length at least k+1. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter λ, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.
ER  -

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