Algebraic structures of MRD codes

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

Algebraic structures of MRD codes

Javier de la Cruz ^1, , Michael Kiermaier ^2, , Alfred Wassermann ^3, and Wolfgang Willems ^4,

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

Abstract
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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, semifields., network coding, quasifields.

Mathematics Subject Classification: Primary: 94B99, 16Y60; Secondary: 15B3.

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.
[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.
[3]	R. H. Bruck and R. C. Bose, The construction of translation planes from projectives spaces,, J. Algebra, 1 (1964), 85.
[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.
[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.
[6]	P. Dembowski, Finite Geometries,, Springer, (1968).
[7]	U. Dempwolff, Semifield planes of order 81,, J. Geom., 89 (): 2008. doi: 10.1007/s00022-008-1995-2.
[8]	E. M. Gabidulin, Theory of codes with maximal rank distance,, Probl. Inform. Transm., 21 (1985), 1.
[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.
[10]	L.-K. Hua, A theorem on matrices over a sfield and its applications,, Acta Math. Sinica, 1 (1951), 109.
[11]	M. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes,, Chapman Hall/CRC, (2007). doi: 10.1201/9781420011142.
[12]	W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103.
[13]	N. Knarr, Quasifields of symplectic translation planes,, J. Combin. Theory Ser. A, 116 (2009), 1080. doi: 10.1016/j.jcta.2008.11.012.
[14]	D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182.
[15]	M. Lavrauw and O. Polverino, Finite semifields,, in Current Research Topocs in Galois Geometry* (eds. J. de Beule and L. Storme)*, (2011).
[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.
[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.
[18]	G. Nebe and W. Willems, On self-dual MRD codes,, Adv. Math. Comm., 10 (2016), 633. doi: 10.3934/amc.2016031.
[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.
[20]	I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements,, Finite Fields Appl., 18 (2012), 1148.
[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.
[22]	R. J. Walker, Determination of division algebras with 32 elements,, Proc. Sympos. Appl. Math., 75 (1962), 83.
[23]	Z.-X. Wan, A proof of the automorphisms of linear groups over a sfield of characteristic 2,, Sci. Sinica, 11 (1962), 1183.
[24]	Z.-X. Wan, Geometry of Matrices,, World Scientific, (1996). doi: 10.1142/9789812830234.
[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.
[26]	H. Zassenhaus, Über endliche Fastkörper,, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187. doi: 10.1007/BF02940723.
[27]	=, ().

show all references

References:

[1]	J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,, Math. Z., 60 (1954), 156.
[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.
[3]	R. H. Bruck and R. C. Bose, The construction of translation planes from projectives spaces,, J. Algebra, 1 (1964), 85.
[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.
[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.
[6]	P. Dembowski, Finite Geometries,, Springer, (1968).
[7]	U. Dempwolff, Semifield planes of order 81,, J. Geom., 89 (): 2008. doi: 10.1007/s00022-008-1995-2.
[8]	E. M. Gabidulin, Theory of codes with maximal rank distance,, Probl. Inform. Transm., 21 (1985), 1.
[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.
[10]	L.-K. Hua, A theorem on matrices over a sfield and its applications,, Acta Math. Sinica, 1 (1951), 109.
[11]	M. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation Planes,, Chapman Hall/CRC, (2007). doi: 10.1201/9781420011142.
[12]	W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103.
[13]	N. Knarr, Quasifields of symplectic translation planes,, J. Combin. Theory Ser. A, 116 (2009), 1080. doi: 10.1016/j.jcta.2008.11.012.
[14]	D. E. Knuth, Finite semifields and projective planes,, J. Algebra, 2 (1965), 182.
[15]	M. Lavrauw and O. Polverino, Finite semifields,, in Current Research Topocs in Galois Geometry* (eds. J. de Beule and L. Storme)*, (2011).
[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.
[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.
[18]	G. Nebe and W. Willems, On self-dual MRD codes,, Adv. Math. Comm., 10 (2016), 633. doi: 10.3934/amc.2016031.
[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.
[20]	I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements,, Finite Fields Appl., 18 (2012), 1148.
[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.
[22]	R. J. Walker, Determination of division algebras with 32 elements,, Proc. Sympos. Appl. Math., 75 (1962), 83.
[23]	Z.-X. Wan, A proof of the automorphisms of linear groups over a sfield of characteristic 2,, Sci. Sinica, 11 (1962), 1183.
[24]	Z.-X. Wan, Geometry of Matrices,, World Scientific, (1996). doi: 10.1142/9789812830234.
[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.
[26]	H. Zassenhaus, Über endliche Fastkörper,, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 187. doi: 10.1007/BF02940723.
[27]	=, ().

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