Algebraic structures of MRD codes

Pages: 499 - 510, Volume 10, Issue 3, August 2016      doi:10.3934/amc.2016021

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Javier de la Cruz - Departamento de Matemáticas, Universidad del Norte, Km 5 Vía Puerto Colombia, Barranquilla, Colombia (email)
Michael Kiermaier - Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany (email)
Alfred Wassermann - Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany (email)
Wolfgang Willems - Department of Mathematics, Otto-von-Guericke-University, 39016 Magdeburg, Germany (email)

Abstract: Based on results in finite geometry we prove the existence of MRD codes in $(\mathbb{F}_q)_{n,n}$ 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.

Keywords:  MRD codes, rank distance, network coding, quasifields, semifields.
Mathematics Subject Classification:  Primary: 94B99, 16Y60; Secondary: 15B33.

Received: April 2015;      Revised: April 2016;      Available Online: August 2016.

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