A new family of linear maximum rank distance codes

In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.

[1]  Rod Gow,et al.  GALOIS EXTENSIONS AND SUBSPACES OF ALTERNATING BILINEAR FORMS WITH SPECIAL RANK PROPERTIES , 2009 .

[2]  P. Dembowski.  Finite geometries , 1997 .

[3]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2008, IEEE Trans. Inf. Theory.

[4]  Norman L. Johnson,et al.  The Collineation Groups of Generalized Twisted Field Planes , 1999 .

[5]  Alberto Ravagnani,et al.  Rank-metric codes and their MacWilliams identities , 2014, ArXiv.

[6]  Thomas Honold,et al.  The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes , 2016, ArXiv.

[7]  Sascha Kurz,et al.  Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4 , 2014 .

[8]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[9]  Katherine Morrison,et al.  Equivalence for Rank-Metric and Matrix Codes and Automorphism Groups of Gabidulin Codes , 2013, IEEE Transactions on Information Theory.

[10]  Ernst M. Gabidulin,et al.  The new construction of rank codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[11]  O. Ore.  On a special class of polynomials , 1933 .

[12]  John Sheekey,et al.  On embeddings of minimum dimension of PG(n,q)×PG(n,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{PG}}(n,q , 2013, Designs, Codes and Cryptography.

[13]  Michel Lavrauw,et al.  A geometric construction of finite semifields , 2007 .

[14]  Rod Gow,et al.  Galois theory and linear algebra , 2009 .

[15]  Daniel Augot,et al.  Rank metric and Gabidulin codes in characteristic zero , 2013, 2013 IEEE International Symposium on Information Theory.

[16]  Ernst M. Gabidulin,et al.  Symmetric Rank Codes , 2004, Probl. Inf. Transm..

[17]  Rocco Trombetti,et al.  Generalized Twisted Gabidulin Codes , 2015, J. Comb. Theory A.

[18]  Michel Lavrauw,et al.  Scattered Spaces in Galois Geometry , 2015, 1512.05251.

[19]  W. Kantor.  Finite semifields , 2005 .

[20]  Guglielmo Lunardon,et al.  Blocking Sets and Derivable Partial Spreads , 2001 .

[21]  Alfred Wassermann,et al.  Algebraic structures of MRD codes , 2015, Adv. Math. Commun..

[22]  Philippe Delsarte,et al.  Bilinear Forms over a Finite Field, with Applications to Coding Theory , 1978, J. Comb. Theory A.

[23]  Giampaolo Menichetti.  On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field , 1977 .

[24]  Kai-Uwe Schmidt,et al.  Symmetric bilinear forms over finite fields with applications to coding theory , 2014, Journal of Algebraic Combinatorics.

[25]  Z. Wan,et al.  Geometry of Matrices , 1996 .

[26]  Thomas Honold,et al.  Poster: A new approach to the Main Problem of Subspace Coding , 2014, 9th International Conference on Communications and Networking in China.

[27]  José Ranilla,et al.  Determination of division algebras with 243 elements , 2010, Finite Fields Their Appl..

[28]  A. Adrian Albert,et al.  Generalized twisted fields. , 1961 .

[29]  Giuseppe Marino,et al.  Non-linear maximum rank distance codes , 2016, Des. Codes Cryptogr..

[30]  Sascha Kurz,et al.  Optimal Binary Subspace Codes of Length 6 , Constant Dimension 3 and Minimum Subspace Distance 4 , 2014 .

[31]  John Sheekey,et al.  Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces , 2014, Electron. J. Comb..

[32]  Giuseppe Marino,et al.  Translation dual of a semifield , 2008, J. Comb. Theory, Ser. A.

[33]  John Sheekey,et al.  Subspaces of matrices with special rank properties , 2010 .

[34]  Thierry P. Berger,et al.  Isometries for rank distance and permutation group of Gabidulin codes , 2003, IEEE Trans. Inf. Theory.

[35]  Michel Lavrauw,et al.  Scattered Spaces with Respect to a Spread in PG(n,q) , 2000 .