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

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.

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

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 -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 . 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 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.

Paper Structure

This paper contains 20 sections, 28 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Background on Linear Codes over Rings
  3. Linear Codes over Finite Fields and Finite Rings
  4. Support and Generator Matrices
  5. Covering and Minimal Codewords
  6. Root Words and Orthogonality
  7. Algebraic and Module-Theoretic Structure of the Galois Ring $\mathrm{GR}(p^{n},\ell)$
  8. Definition and Basic Structure
  9. $p$-Adic Representation and Teichmüller System
  10. Zero Divisors and Ideal Structure
  11. Module Structure and Root Words
  12. Frobenius Property
  13. Minimality Criteria and Structural Parameters over $\mathrm{GR}(p^{n},\ell)$
  14. Necessary and Sufficient Conditions and Parameters of a Minimal Linear Code over a Galois Ring
  15. Parameters of Minimal Linear Codes over $\mathrm{GR}(p^{n},\ell)$
  16. ...and 5 more sections

Key Result

Proposition 8

Let $v \in \mathrm{GR}(p^{n},\ell)^{m}$ be a root word. Then the orthogonal module is a free $\mathrm{GR}(p^{n},\ell)$-module of rank $m-1$.

Theorems & Definitions (66)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • Proposition 9
  • Remark 10
  • ...and 56 more