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Minimum Schubert Codewords and Second-Minimum Grassmann Codewords

  • Muskan Khaneja
  • Prasant Singh
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Abstract

In this paper, we give a classification of the minimum weight codewords of Schubert codes Cα(ℓ, m) by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of q and all α. We use this classification to prove that a codeword of the Grassmann code C(ℓ, m) has the second minimum weight if and only if it is indexed by an element of bigwedgem−ℓV that can be written as the product of a decomposable (m−ℓ−2)-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.

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Cite

BibTeXRIS
BibTeX
@misc{khaneja2026minimum,
  title = {{Minimum Schubert Codewords and Second-Minimum Grassmann Codewords}},
  author = {Muskan Khaneja and Prasant Singh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04916},
  eprint = {2609.04916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04916},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Minimum Schubert Codewords and Second-Minimum Grassmann Codewords
AU  - Muskan Khaneja
AU  - Prasant Singh
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.04916
UR  - https://arxiv.org/abs/2609.04916
AB  - In this paper, we give a classification of the minimum weight codewords of Schubert codes C_α(ℓ, m) by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of q and all α. We use this classification to prove that a codeword of the Grassmann code C(ℓ, m) has the second minimum weight if and only if it is indexed by an element of bigwedge^m−ℓV that can be written as the product of a decomposable (m−ℓ−2)-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.
ER  -