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Nonexistence results and constructions for signed difference sets

  • Zhiwen He
  • Jiayan Wu
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Abstract

Signed difference sets (SDSs) extend ordinary difference sets by allowing negative coefficients. We establish new nonexistence criteria and existence constructions for SDSs in finite abelian groups. Using the classical self-conjugate-prime method and a support-refined quotient method, we derive four obstructions, including the C2-quotient, near-full-support, and small-defect obstructions. Applied cumulatively to the 67,823 open group-specific cases in the database with 9≤ v≤499, these criteria rule out 45,361 cases. For existence, we construct an infinite family from PCP-type regular partial difference sets arising from Desarguesian spreads and an explicit $(125,28,3)$-SDS in C53 using quartic multiplicative characters. These constructions settle three further open cases.

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Cite

BibTeXRIS
BibTeX
@misc{he2026nonexistence,
  title = {{Nonexistence results and constructions for signed difference sets}},
  author = {Zhiwen He and Jiayan Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.22936},
  eprint = {2608.22936},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22936},
}
RIS
TY  - UNPB
TI  - Nonexistence results and constructions for signed difference sets
AU  - Zhiwen He
AU  - Jiayan Wu
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.22936
UR  - https://arxiv.org/abs/2608.22936
AB  - Signed difference sets (SDSs) extend ordinary difference sets by allowing negative coefficients. We establish new nonexistence criteria and existence constructions for SDSs in finite abelian groups. Using the classical self-conjugate-prime method and a support-refined quotient method, we derive four obstructions, including the C_2-quotient, near-full-support, and small-defect obstructions. Applied cumulatively to the 67,823 open group-specific cases in the database with 9≤ v≤499, these criteria rule out 45,361 cases. For existence, we construct an infinite family from PCP-type regular partial difference sets arising from Desarguesian spreads and an explicit $(125,28,3)$-SDS in C_5^3 using quartic multiplicative characters. These constructions settle three further open cases.
ER  -