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PreprintarXiv licencearXiv

The Last Seven Open Radii for Perfect Codes in the Johnson Scheme

  • Xiande Zhang
  • Wenjie Zhong
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Abstract

Delsarte (1973) conjectured that there are no nontrivial perfect codes in the Johnson scheme. In this paper, we prove that there are no nontrivial e-perfect codes in the Johnson scheme for e∈{1,2,4,9,10,12,16}. This paper complements "Perfect Codes in the Johnson Scheme Hardly Exist", thus proving Delsarte's conjecture completely.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026last,
  title = {{The Last Seven Open Radii for Perfect Codes in the Johnson Scheme}},
  author = {Xiande Zhang and Wenjie Zhong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23368},
  eprint = {2609.23368},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23368},
}
RIS
TY  - UNPB
TI  - The Last Seven Open Radii for Perfect Codes in the Johnson Scheme
AU  - Xiande Zhang
AU  - Wenjie Zhong
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23368
UR  - https://arxiv.org/abs/2609.23368
AB  - Delsarte (1973) conjectured that there are no nontrivial perfect codes in the Johnson scheme. In this paper, we prove that there are no nontrivial e-perfect codes in the Johnson scheme for e∈{1,2,4,9,10,12,16}. This paper complements "Perfect Codes in the Johnson Scheme Hardly Exist", thus proving Delsarte's conjecture completely.
ER  -