Function-Based Minimal Linear Codes over Galois Rings $\mathrm{GR}(p^{n}, \ell)$: Minimality Criteria and Infinite Constructions
2026-03-27 • Information Theory
Information Theory
AI summaryⓘ
The authors extend previous work by providing clear rules to identify special types of linear codes, called minimal codes, but now over more complicated number systems known as Galois rings instead of simpler fields. They handle new algebraic difficulties due to zero divisors and richer structures in these rings. Using advanced concepts like Frobenius duality, they give exact conditions for when codes made from functions over Galois rings are minimal. They also build large new families of such codes, expanding on earlier constructions by Wu and colleagues. This work ties together ideas from coding theory, ring theory, and function-based code design.
Minimal linear codesGalois ringsFrobenius dualityZero divisorsModule theoryFunction-based minimalityLinear codesCryptographyCode length boundsRing theory
Authors
Biplab Chatterjee, Sihem Mesnager, Ratnesh Kumar Mishra, Makhan Maji, Kalyan Hansda
Abstract
In this paper, we extend a necessary and sufficient condition for a linear code over a Galois ring to be minimal and establish new bounds on the length of an $m$-dimensional minimal linear code. Building upon this structural characterization, we further generalize the function-based minimality criteria introduced by Wu \emph{et al.} (Cryptogr. Commun. 14, 875-895, 2022) from the finite field setting to the framework of Galois rings. The transition from fields to rings introduces substantial algebraic challenges due to the presence of zero divisors and the richer module structure of $\mathrm{GR}(p^{n},\ell)$. By exploiting Frobenius duality and the chain structure of Galois rings, we derive refined necessary and sufficient conditions ensuring that linear codes arising from functions over $\mathrm{GR}(p^{n},\ell)$ are minimal. As an application of these criteria, we construct several infinite families of minimal linear codes over Galois rings, thereby significantly generalizing the constructions of Wu \emph{et al.} to the ring setting. Our results provide a unified framework that connects minimality theory, module duality over Frobenius rings, and function-based code constructions.