Infinite families of 3-designs from linear and nonlinear codes

Abstract

The connection between coding theory and combinatorial t-designs is an important research topic at the intersection of coding theory and combinatorics. Let q=p^m, where p is an odd prime and m≥ 2. In this paper, we investigate a class of linear codes 𝒞 over 𝔽_q² and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of 𝒞 and 𝒞^⊥, we show that the supports of the codewords of every nonzero weight in 𝒞 and {the supports of the codewords of weight $4$ in 𝒞^⊥} form $3$-designs. We further investigate a class of nonlinear codes 𝒞₂ associated with 𝒞, and prove that the supports of the codewords of every nonzero Hamming weight in 𝒞₂ also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from 𝒞^⊥ an entanglement-assisted quantum error-correcting code with parameters [[q+1,q−3,4;4]]_q. We also prove that 𝒞 is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code 𝒞 {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.

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BibTeX RIS

BibTeX

@misc{xiong2026infinite,
  title = {{Infinite families of 3-designs from linear and nonlinear codes}},
  author = {Shiyan Xiong and Xiaoqiang Wang and Dabin Zheng and Qinqin Ji},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15525},
  eprint = {2609.15525},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15525},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Infinite families of 3-designs from linear and nonlinear codes
AU  - Shiyan Xiong
AU  - Xiaoqiang Wang
AU  - Dabin Zheng
AU  - Qinqin Ji
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15525
UR  - https://arxiv.org/abs/2609.15525
AB  - The connection between coding theory and combinatorial t-designs is an important research topic at the intersection of coding theory and combinatorics. Let q=p^m, where p is an odd prime and m≥ 2. In this paper, we investigate a class of linear codes 𝒞 over 𝔽_q^2 and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of 𝒞 and 𝒞^⊥, we show that the supports of the codewords of every nonzero weight in 𝒞 and {the supports of the codewords of weight $4$ in 𝒞^⊥} form $3$-designs. We further investigate a class of nonlinear codes 𝒞_2 associated with 𝒞, and prove that the supports of the codewords of every nonzero Hamming weight in 𝒞_2 also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from 𝒞^⊥ an entanglement-assisted quantum error-correcting code with parameters [[q+1,q−3,4;4]]_q. We also prove that 𝒞 is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code 𝒞 {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.
ER  -

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