Graph-Based Linear Codes Associated with the Finite Ring $\mathbb{F}_p[x]/\langle x^4 \rangle$

2026-10-06

AI Summary

This study investigates the parameters and constructions of linear codes induced by zero-divisor graphs over finite rings. Methodologically, it pioneers the integration of equitable partitions of zero-divisor graphs with MacWilliams identities, employing graph-theoretic, algebraic coding, and automorphism group analysis techniques to parameterize adjacency and Laplacian codes over $\mathbb{F}_p$ and derive their weight distributions. The principal contributions are threefold: determining the parameters of binary incidence codes and constructing induced bipartite subgraph codes with dual minimum distance four; proving that the constructed codes possess the linear complementary dual (LCD) property; and successfully establishing two families of Griesmer-optimal linear codes, including constant-weight minimal codes. Collectively, this work achieves a graph-based code construction that simultaneously exhibits specific dual distances and the LCD property.

Abstract

We investigate the linear codes derived from the zero-divisor graph $Γ(R)$ of the finite local ring $R = \mathbb{F}_p[x]/\langle x^4 \rangle$ 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 $\mathbb{F}_p$, 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 $\mathbb{F}_p$, comprising a constant-weight minimal code and an optimal two-weight code.

Problem

• zero-divisor graph
• linear codes
• finite local ring
• Laplacian code
• Griesmer-optimal codes

Innovation

• zero-divisor graph
• linear codes
• Laplacian code
• Griesmer-optimal
• LCD property