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PreprintCC BY 4.0arXiv

Four-Entropic Matroids Are Quaternary

  • Mohammad Hossein Kalantari
  • Shahram Khazaei
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Abstract

For an integer q≥2, a matroid is q-entropic if its rank function, multiplied by log q, is the joint-entropy function of random variables on a q-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over F4. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal F4-linear scheme.

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Cite

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BibTeX
@misc{kalantari2026fourentropic,
  title = {{Four-Entropic Matroids Are Quaternary}},
  author = {Mohammad Hossein Kalantari and Shahram Khazaei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.20553},
  eprint = {2608.20553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20553},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Four-Entropic Matroids Are Quaternary
AU  - Mohammad Hossein Kalantari
AU  - Shahram Khazaei
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.20553
UR  - https://arxiv.org/abs/2608.20553
AB  - For an integer q≥2, a matroid is q-entropic if its rank function, multiplied by log q, is the joint-entropy function of random variables on a q-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over F_4. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal F_4-linear scheme.
ER  -