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Algebraic structures of MRD Codes

Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems

TL;DR

The work investigates MRD codes in $(\mathbb{F}_q)_{m,n}$, focusing on cases where the minimum distance equals $n$ and highlighting codes that are not Gabidulin. It establishes a deep algebraic correspondence for $k=1$, $n=m$: MRD codes arise from spreadsets that equip $W=\mathbb{F}_q^n$ with a finite quasifield (and semifield or division algebra when additive/linear), with equivalence of codes controlled by isotopy of the corresponding algebras (including transpose). It further shows that symmetric MRD codes correspond to invariant symmetric bilinear forms on the quasifield and are, in many settings, equivalent to Gabidulin codes, while finite nearfields and exceptional constructions yield non-Gabidulin MRD codes, including explicit examples and higher-$k$ phenomena demonstrated computationally. Together, these results provide a unifying algebraic framework linking MRD codes to spreadsets, quasifields, semifields, division algebras, and nearfields, revealing rich families of MRD codes beyond Gabidulin and informing code equivalence via isotopy.

Abstract

Based on results in finite geometry we prove the existence of MRD codes in (F_q)_(n,n) with minimum distance n which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.

Algebraic structures of MRD Codes

TL;DR

The work investigates MRD codes in , focusing on cases where the minimum distance equals and highlighting codes that are not Gabidulin. It establishes a deep algebraic correspondence for , : MRD codes arise from spreadsets that equip with a finite quasifield (and semifield or division algebra when additive/linear), with equivalence of codes controlled by isotopy of the corresponding algebras (including transpose). It further shows that symmetric MRD codes correspond to invariant symmetric bilinear forms on the quasifield and are, in many settings, equivalent to Gabidulin codes, while finite nearfields and exceptional constructions yield non-Gabidulin MRD codes, including explicit examples and higher- phenomena demonstrated computationally. Together, these results provide a unifying algebraic framework linking MRD codes to spreadsets, quasifields, semifields, division algebras, and nearfields, revealing rich families of MRD codes beyond Gabidulin and informing code equivalence via isotopy.

Abstract

Based on results in finite geometry we prove the existence of MRD codes in (F_q)_(n,n) with minimum distance n which are essentially different from Gabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.

Paper Structure

This paper contains 6 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. On the structure of MRD codes with $k=1$ and $n=m$
  3. Isotopy and equivalence
  4. Symmetric MRD codes
  5. Finite nearfields
  6. MRD codes with $k>1$ and $n=m$

Key Result

Theorem 1

(Hua, Wan) If $\varphi$ is an isometry of $(\mathbb{F}_q)_{m,n}$ with $m,n \geq 2$, then there exist matrices $X \in \hbox{\rm GL}(m,q), Y \in \hbox{\rm GL}(n,q)$ and $Z \in (F_q)_{m,n}$ such that where $\sigma$ is an automorphism of the field $\mathbb{F}_q$ acting on the entries of $A$, or, but only in case $m=n$, where $A^t$ denotes the transpose of $A$. If $\varphi$ is additive, then obviousl

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • ...and 11 more