August  2016, 10(3): 499-510. doi: 10.3934/amc.2016021

Algebraic structures of MRD codes

1. 

Departamento de Matemáticas, Universidad del Norte, Km 5 Vía Puerto Colombia, Barranquilla, Colombia

2. 

Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth

3. 

Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

4. 

Department of Mathematics, Otto-von-Guericke-University, 39016 Magdeburg

Received  April 2015 Revised  April 2016 Published  August 2016

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.
Citation: Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021
References:
[1]

J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,, Math. Z., 60 (1954), 156.   Google Scholar

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W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

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R. H. Bruck and R. C. Bose, The construction of translation planes from projectives spaces,, J. Algebra, 1 (1964), 85.   Google Scholar

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M. Cordero and G. P. Wene, A survey of finite semifields,, Discrete Math., 208/209 (1999), 125.  doi: 10.1016/S0012-365X(99)00068-0.  Google Scholar

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P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Combin. Theory Ser. A, 25 (1978), 226.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[6]

P. Dembowski, Finite Geometries,, Springer, (1968).   Google Scholar

[7]

U. Dempwolff, Semifield planes of order 81,, J. Geom., 89 (): 2008.  doi: 10.1007/s00022-008-1995-2.  Google Scholar

[8]

E. M. Gabidulin, Theory of codes with maximal rank distance,, Probl. Inform. Transm., 21 (1985), 1.   Google Scholar

[9]

E. M. Gabidulin and N. I. Pilipchuk, Symmetric matrices and codes correcting rank errors beyond the $\lfloor \frac{d-1}{2} \rfloor$ bound,, Discrete Appl. Math., 154 (2006), 305.  doi: 10.1016/j.dam.2005.03.012.  Google Scholar

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L.-K. Hua, A theorem on matrices over a sfield and its applications,, Acta Math. Sinica, 1 (1951), 109.   Google Scholar

[11]

M. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes,, Chapman Hall/CRC, (2007).  doi: 10.1201/9781420011142.  Google Scholar

[12]

W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103.   Google Scholar

[13]

N. Knarr, Quasifields of symplectic translation planes,, J. Combin. Theory Ser. A, 116 (2009), 1080.  doi: 10.1016/j.jcta.2008.11.012.  Google Scholar

[14]

D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182.   Google Scholar

[15]

M. Lavrauw and O. Polverino, Finite semifields,, in Current Research Topocs in Galois Geometry (eds. J. de Beule and L. Storme), (2011).   Google Scholar

[16]

G. Marino and O. Polverino, On isotopisms and strong isotopisms of commutative presemifields,, J. Algebr. Combin., 36 (2012), 247.  doi: 10.1007/s10801-011-0334-0.  Google Scholar

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K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes,, IEEE Trans. Inform. Theory, 60 (2014), 7035.  doi: 10.1109/TIT.2014.2359198.  Google Scholar

[18]

G. Nebe and W. Willems, On self-dual MRD codes,, Adv. Math. Comm., 10 (2016), 633.  doi: 10.3934/amc.2016031.  Google Scholar

[19]

I. F. Rúa, E. F. Combarro and J. Ranilla, Classification of semifields of order 64,, J. Algebra, 322 (2009), 4011.  doi: 10.1016/j.jalgebra.2009.02.020.  Google Scholar

[20]

I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements,, Finite Fields Appl., 18 (2012), 1148.   Google Scholar

[21]

K.-U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory,, J. Algebr. Combin., 42 (2015), 635.  doi: 10.1007/s10801-015-0595-0.  Google Scholar

[22]

R. J. Walker, Determination of division algebras with 32 elements,, Proc. Sympos. Appl. Math., 75 (1962), 83.   Google Scholar

[23]

Z.-X. Wan, A proof of the automorphisms of linear groups over a sfield of characteristic 2,, Sci. Sinica, 11 (1962), 1183.   Google Scholar

[24]

Z.-X. Wan, Geometry of Matrices,, World Scientific, (1996).  doi: 10.1142/9789812830234.  Google Scholar

[25]

S. Yang and T. Honold, Good random matrices over finite fields,, Adv. Math. Commun., 6 (2012), 203.  doi: 10.3934/amc.2012.6.203.  Google Scholar

[26]

H. Zassenhaus, Über endliche Fastkörper,, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187.  doi: 10.1007/BF02940723.  Google Scholar

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=, ().   Google Scholar

show all references

References:
[1]

J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,, Math. Z., 60 (1954), 156.   Google Scholar

[2]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

R. H. Bruck and R. C. Bose, The construction of translation planes from projectives spaces,, J. Algebra, 1 (1964), 85.   Google Scholar

[4]

M. Cordero and G. P. Wene, A survey of finite semifields,, Discrete Math., 208/209 (1999), 125.  doi: 10.1016/S0012-365X(99)00068-0.  Google Scholar

[5]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory,, J. Combin. Theory Ser. A, 25 (1978), 226.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[6]

P. Dembowski, Finite Geometries,, Springer, (1968).   Google Scholar

[7]

U. Dempwolff, Semifield planes of order 81,, J. Geom., 89 (): 2008.  doi: 10.1007/s00022-008-1995-2.  Google Scholar

[8]

E. M. Gabidulin, Theory of codes with maximal rank distance,, Probl. Inform. Transm., 21 (1985), 1.   Google Scholar

[9]

E. M. Gabidulin and N. I. Pilipchuk, Symmetric matrices and codes correcting rank errors beyond the $\lfloor \frac{d-1}{2} \rfloor$ bound,, Discrete Appl. Math., 154 (2006), 305.  doi: 10.1016/j.dam.2005.03.012.  Google Scholar

[10]

L.-K. Hua, A theorem on matrices over a sfield and its applications,, Acta Math. Sinica, 1 (1951), 109.   Google Scholar

[11]

M. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes,, Chapman Hall/CRC, (2007).  doi: 10.1201/9781420011142.  Google Scholar

[12]

W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103.   Google Scholar

[13]

N. Knarr, Quasifields of symplectic translation planes,, J. Combin. Theory Ser. A, 116 (2009), 1080.  doi: 10.1016/j.jcta.2008.11.012.  Google Scholar

[14]

D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182.   Google Scholar

[15]

M. Lavrauw and O. Polverino, Finite semifields,, in Current Research Topocs in Galois Geometry (eds. J. de Beule and L. Storme), (2011).   Google Scholar

[16]

G. Marino and O. Polverino, On isotopisms and strong isotopisms of commutative presemifields,, J. Algebr. Combin., 36 (2012), 247.  doi: 10.1007/s10801-011-0334-0.  Google Scholar

[17]

K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes,, IEEE Trans. Inform. Theory, 60 (2014), 7035.  doi: 10.1109/TIT.2014.2359198.  Google Scholar

[18]

G. Nebe and W. Willems, On self-dual MRD codes,, Adv. Math. Comm., 10 (2016), 633.  doi: 10.3934/amc.2016031.  Google Scholar

[19]

I. F. Rúa, E. F. Combarro and J. Ranilla, Classification of semifields of order 64,, J. Algebra, 322 (2009), 4011.  doi: 10.1016/j.jalgebra.2009.02.020.  Google Scholar

[20]

I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements,, Finite Fields Appl., 18 (2012), 1148.   Google Scholar

[21]

K.-U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory,, J. Algebr. Combin., 42 (2015), 635.  doi: 10.1007/s10801-015-0595-0.  Google Scholar

[22]

R. J. Walker, Determination of division algebras with 32 elements,, Proc. Sympos. Appl. Math., 75 (1962), 83.   Google Scholar

[23]

Z.-X. Wan, A proof of the automorphisms of linear groups over a sfield of characteristic 2,, Sci. Sinica, 11 (1962), 1183.   Google Scholar

[24]

Z.-X. Wan, Geometry of Matrices,, World Scientific, (1996).  doi: 10.1142/9789812830234.  Google Scholar

[25]

S. Yang and T. Honold, Good random matrices over finite fields,, Adv. Math. Commun., 6 (2012), 203.  doi: 10.3934/amc.2012.6.203.  Google Scholar

[26]

H. Zassenhaus, Über endliche Fastkörper,, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187.  doi: 10.1007/BF02940723.  Google Scholar

[27]

=, ().   Google Scholar

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