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On the existence of MRD self-dual codes

Grégory Berhuy

TL;DR

This work investigates the existence of self-dual MRD codes in the Gabidulin (rank-metric) framework and analyzes their relationship to Delsarte self-duality. It proves nonexistence of MRD self-dual codes in odd characteristic for Lagrangian codes and establishes a precise criterion in the finite-field case $m=n$: a self-dual MRD code exists iff $q\equiv n\equiv 3\pmod{4}$; in characteristic two, no such codes exist for the standard inner product, but a Lagrangian analogue can be constructed. The authors connect Gabidulin and Delsarte dualities via dual bases and trace forms, showing transfer under suitable orthonormal bases, and they provide explicit constructions in characteristic two that yield Lagrangian MRD codes over finite fields. Overall, the paper clarifies when self-dual MRD codes can exist, relates two duality notions, and offers a robust framework for extending self-duality concepts to characteristic two. The results have implications for the structure and construction of rank-metric codes in both theoretical and finite-field applications.

Abstract

In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and $F=\mathbb{F}_q$, a self-dual MRD code exists if and only if $q\equiv n\equiv 3 \ [4].$

On the existence of MRD self-dual codes

TL;DR

This work investigates the existence of self-dual MRD codes in the Gabidulin (rank-metric) framework and analyzes their relationship to Delsarte self-duality. It proves nonexistence of MRD self-dual codes in odd characteristic for Lagrangian codes and establishes a precise criterion in the finite-field case : a self-dual MRD code exists iff ; in characteristic two, no such codes exist for the standard inner product, but a Lagrangian analogue can be constructed. The authors connect Gabidulin and Delsarte dualities via dual bases and trace forms, showing transfer under suitable orthonormal bases, and they provide explicit constructions in characteristic two that yield Lagrangian MRD codes over finite fields. Overall, the paper clarifies when self-dual MRD codes can exist, relates two duality notions, and offers a robust framework for extending self-duality concepts to characteristic two. The results have implications for the structure and construction of rank-metric codes in both theoretical and finite-field applications.

Abstract

In this paper, we investigate the existence of self-dual MRD codes , where is an arbitrary field extension of degree . We then apply our results to the case of finite fields, and prove that if and , a self-dual MRD code exists if and only if
Paper Structure (7 sections, 17 theorems, 29 equations)

This paper contains 7 sections, 17 theorems, 29 equations.

Table of Contents

  1. Basic definitions and summary of the main results
  2. Preliminaries on the rank distance
  3. Lagrangian codes
  4. Self-dual MRD codes
  5. Gabidulin self-duality versus Delsarte self-duality
  6. The case of characteristic two revisited
  7. Conclusion

Key Result

Lemma 1

For any $\mathbf{c}\in L^n,$$M_{\boldsymbol{\alpha}}(\mathbf{c})$ is the unique matrix $M\in {\rm M}_{m\times n}(F)$ satisfying $\mathbf{c}=\boldsymbol{\alpha} M.$

Theorems & Definitions (37)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Proposition 1
  • ...and 27 more