Function-Based Minimal Linear Codes over Galois Rings $\mathrm{GR}(p^{n}, \ell)$: Minimality Criteria and Infinite Constructions

2026-03-27

AI Summary

This work investigates the construction of infinite families of minimal linear codes over Galois rings containing zero divisors and establishes a general criterion for minimality. Addressing the algebraic challenges posed by zero divisors and the module structure inherent in Galois rings, the study extends the function-based theory of minimal codes—originally developed over finite fields—to the setting of Galois rings. By leveraging Frobenius duality, the chain structure of the ring, and tools from module theory, the authors derive necessary and sufficient conditions for a linear code to be minimal, establish lower bounds on code length, and successfully construct several infinite families of minimal linear codes. This contribution provides a unified framework for determining minimality in the context of Galois rings.

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.

Problem

• minimal linear codes
• Galois rings
• function-based construction
• zero divisors
• Frobenius duality

Innovation

• minimal linear codes
• Galois rings
• function-based construction
• Frobenius duality
• zero divisors