Graph codes from finite ring produce optimal weighted linear codes
Graph-Based Linear Codes Associated with the Finite Ring $\mathbb{F}_p[x]/\langle x^4 \rangle$
Information Theory
Summary
This paper looks at special kinds of codes made from a specific math object called a finite local ring. The authors study the connections in a graph built from this ring to find new codes with good error-detection and correction features. They figure out key properties like code length, weight, and symmetry. Using patterns in this graph, they create two strong types of codes that are minimal or have only two weights, which are useful for communication systems.
What this means in practice
- •For error correction engineers: Design linear codes with guaranteed dual minimum distance and known weight distributions for reliable data transmission systems.
- •For network security developers: Use constant-weight minimal codes from the finite ring graph constructions to build secure communication protocols resistant to certain attacks.
Authors
Apurba Sarkar, Kalyan Hansda, Makhan Maji
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.