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Optimal Rank-Metric Codes with Rank-Locality from Drinfeld Modules

Luca Bastioni, Mohamed O. Darwish, Giacomo Micheli

TL;DR

The work introduces a novel approach to constructing optimal rank-metric codes with rank-locality by leveraging Drinfeld modules over finite fields and Dirichlet-type polynomial arithmetic progressions. The general construction encodes messages through $q$-linearized polynomials composed with Drinfeld module actions, producing codes with length proportional to the sum of local blocks and proven locality properties. Supersingular and rank-1 variants provide ambient-field control, enabling explicit realizations and infinite families of codes that meet rank-locality bounds. The results significantly expand the parameter space for locally recoverable rank-metric codes and offer a framework blending algebraic function-field theory with coding theory for practical distributed storage and cryptographic applications.

Abstract

We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family of optimal rank-metric codes with rank-locality, i.e. every code in our family achieves the information theoretical bound for rank-metric codes with rank-locality.

Optimal Rank-Metric Codes with Rank-Locality from Drinfeld Modules

TL;DR

The work introduces a novel approach to constructing optimal rank-metric codes with rank-locality by leveraging Drinfeld modules over finite fields and Dirichlet-type polynomial arithmetic progressions. The general construction encodes messages through -linearized polynomials composed with Drinfeld module actions, producing codes with length proportional to the sum of local blocks and proven locality properties. Supersingular and rank-1 variants provide ambient-field control, enabling explicit realizations and infinite families of codes that meet rank-locality bounds. The results significantly expand the parameter space for locally recoverable rank-metric codes and offer a framework blending algebraic function-field theory with coding theory for practical distributed storage and cryptographic applications.

Abstract

We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family of optimal rank-metric codes with rank-locality, i.e. every code in our family achieves the information theoretical bound for rank-metric codes with rank-locality.
Paper Structure (16 sections, 19 theorems, 82 equations, 1 table)

This paper contains 16 sections, 19 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Drinfeld Modules
  4. Rank Metric Codes
  5. General Theorem and Main Construction
  6. Main Code Construction
  7. Supersingular Drinfeld Modules
  8. Supersingular Drinfeld modules of rank $R>1$
  9. Drinfeld modules of rank $R=1$
  10. Examples
  11. Explicit Construction via a Carlitz module
  12. Explicit Construction via a supersingular Drinfeld module
  13. Choosing the polynomial $\mathfrak{p}$
  14. An Infinite Family of Optimal Rank Metric Codes with Locality
  15. $L$-series and Dirichlet Theorem
  16. ...and 1 more sections

Key Result

Proposition 2.10

Papikian2023 If $\operatorname{char}_A(K) = \mathfrak{p} \neq 0$ and $\phi$ is a Drinfeld module of rank $R$ over $K$, then there is an integer $1 \leq H (\phi) \leq R$, called the height of $\phi$, such that for all $0\neq a\in A$ we have where $\mathop{\mathrm{ord}}\nolimits_\mathfrak{p}(a)$ is the largest power of $\mathfrak{p}$ dividing $a$.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 41 more