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Further results on the classification of MDS codes
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 |
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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