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PreprintCC BY 4.0arXiv

Graph-Based Linear Codes Associated with the Finite Ring 𝔽p[x]/⟨ x4 ⟩

  • Apurba Sarkar
  • Kalyan Hansda
  • Makhan Maji
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Abstract

We investigate the linear codes derived from the zero-divisor graph Γ(R) of the finite local ring R = 𝔽p[x]/⟨ x4 ⟩ for an odd prime p. Using an equitable partition of Z*(R), we determine the parameters of the binary incidence code of Γ(R) and construct induced bipartite subgraph codes achieving dual minimum distance $4$. Over 𝔽p, we parameterize the adjacency and Laplacian codes, derive the closed-form weight distribution of the Laplacian code via the MacWilliams identity, and establish its LCD property. Additionally, we determine the automorphism groups of the graph and its matrix codes. Finally, from extremal subgraphs of Γ(R), we construct two families of Griesmer-optimal linear codes over 𝔽p, comprising a constant-weight minimal code and an optimal two-weight code.

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Cite

BibTeXRIS
BibTeX
@misc{sarkar2026graphbased,
  title = {{Graph-Based Linear Codes Associated with the Finite Ring $\mathbbF_p[x]/\langle x^4 \rangle$}},
  author = {Apurba Sarkar and Kalyan Hansda and Makhan Maji},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08480},
  eprint = {2610.08480},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08480},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Graph-Based Linear Codes Associated with the Finite Ring 𝔽_p[x]/⟨ x^4 ⟩
AU  - Apurba Sarkar
AU  - Kalyan Hansda
AU  - Makhan Maji
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08480
UR  - https://arxiv.org/abs/2610.08480
AB  - We investigate the linear codes derived from the zero-divisor graph Γ(R) of the finite local ring R = 𝔽_p[x]/⟨ x^4 ⟩ for an odd prime p. Using an equitable partition of Z^*(R), we determine the parameters of the binary incidence code of Γ(R) and construct induced bipartite subgraph codes achieving dual minimum distance $4$. Over 𝔽_p, we parameterize the adjacency and Laplacian codes, derive the closed-form weight distribution of the Laplacian code via the MacWilliams identity, and establish its LCD property. Additionally, we determine the automorphism groups of the graph and its matrix codes. Finally, from extremal subgraphs of Γ(R), we construct two families of Griesmer-optimal linear codes over 𝔽_p, comprising a constant-weight minimal code and an optimal two-weight code.
ER  -