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External Difference Families Arising from Two or Three Cyclotomic Classes

  • Miwako Mishima
  • Yu Tsunoda
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Abstract

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime p=12k+1, the family {C06,C26,C46} forms an EDF in 𝔽p if and only if k is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.

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Cite

BibTeXRIS
BibTeX
@misc{mishima2026external,
  title = {{External Difference Families Arising from Two or Three Cyclotomic Classes}},
  author = {Miwako Mishima and Yu Tsunoda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07266},
  eprint = {2609.07266},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07266},
}
RIS
TY  - UNPB
TI  - External Difference Families Arising from Two or Three Cyclotomic Classes
AU  - Miwako Mishima
AU  - Yu Tsunoda
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07266
UR  - https://arxiv.org/abs/2609.07266
AB  - We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime p=12k+1, the family {C_0^6,C_2^6,C_4^6} forms an EDF in 𝔽_p if and only if k is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.
ER  -