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Geometric difference families and related variable-weight geometric orthogonal codes

  • Huizhen Gao
  • Yichen Bai
  • Xiaowei Su
  • Zihong Tian
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Abstract

Variable-weight geometric orthogonal codes (GOCs) were introduced by Doty and Winslow for designing the macrobonds in 3D DNA origami. Recently, Wang et al. introduced the concept of geometric difference families (GDFs) and established the equivalence between GOCs and a certain type of GDFs. Based on the relationship, Su et al. gave two classes of variable-weight GOCs. In this paper, we first prove the existence of (n*m,K,1)-GDFs for K ={3,5} and {3,6}. For further research, we completely determine the existence of all the remaining parameters for an (n*m,{3,4,5},1)-GDF by Su et al. As consequences, the perfect (n*m,K,1)-GOCs for K = {3,5}, {3,6} and {3,4,5} are obtained.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026geometric,
  title = {{Geometric difference families and related variable-weight geometric orthogonal codes}},
  author = {Huizhen Gao and Yichen Bai and Xiaowei Su and Zihong Tian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29202},
  eprint = {2609.29202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29202},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Geometric difference families and related variable-weight geometric orthogonal codes
AU  - Huizhen Gao
AU  - Yichen Bai
AU  - Xiaowei Su
AU  - Zihong Tian
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29202
UR  - https://arxiv.org/abs/2609.29202
AB  - Variable-weight geometric orthogonal codes (GOCs) were introduced by Doty and Winslow for designing the macrobonds in 3D DNA origami. Recently, Wang et al. introduced the concept of geometric difference families (GDFs) and established the equivalence between GOCs and a certain type of GDFs. Based on the relationship, Su et al. gave two classes of variable-weight GOCs. In this paper, we first prove the existence of (n*m,K,1)-GDFs for K ={3,5} and {3,6}. For further research, we completely determine the existence of all the remaining parameters for an (n*m,{3,4,5},1)-GDF by Su et al. As consequences, the perfect (n*m,K,1)-GOCs for K = {3,5}, {3,6} and {3,4,5} are obtained.
ER  -