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A new family of linear maximum rank distance codes
1. | School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland |
References:
[1] |
J. Ai, T. Honold and H. Liu, The expurgation-augmentation method for constructing good plane subspace codes, preprint, arXiv:1601.01502 |
[2] |
A. A. Albert, Generalized twisted fields, Pacific J. Math., 11 (1961), 1-8. |
[3] |
D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in Proc. ISIT 2013, 509-513. |
[4] |
S. Ball, G. Ebert and M. Lavrauw, A geometric construction of finite semifields, J. Algebra, 311 (2007), 117-129.
doi: 10.1016/j.jalgebra.2006.11.044. |
[5] |
T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, IEEE Trans. Inform. Theory, 49 (2003), 3016-3019.
doi: 10.1109/TIT.2003.819322. |
[6] |
M. Biliotti, V. Jha and N. L. Johnson, The collineation groups of generalized twisted field planes, Geom. Dedicata, 76 (1999), 97-126.
doi: 10.1023/A:1005089016092. |
[7] |
A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $\PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243.
doi: 10.1023/A:1005283806897. |
[8] |
A. Cossidente, G. Marino and F. Pavese, Non-linear maximum rank distance codes, preprint.
doi: 10.1007/s10623-015-0108-0. |
[9] |
J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes, Adv. Math. Commun., 10 (2016), 499-510.
doi: 10.3934/amc.2016021. |
[10] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] | |
[12] |
U. Dempwolff, Translation Planes of Small Order, http://www.mathematik.uni-kl.de/~dempw/dempw_Plane.html |
[13] |
J.-G. Dumas, R. Gow, G. McGuire and J. Sheekey, Subspaces of matrices with special rank properties, Linear Algebra Appl., 433 (2010), 191-202.
doi: 10.1016/j.laa.2010.02.015. |
[14] |
E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. |
[15] |
E. Gabidulin and A. Kshevetskiy, The new construction of rank codes, in Proc. ISIT 2005. |
[16] |
E. M. Gabidulin and N. I. Pilipchuk, Symmetric rank codes, Probl. Inf. Transm., 40 (2004), 103-117.
doi: 10.1023/B:PRIT.0000043925.67309.c6. |
[17] |
M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[18] |
R. Gow, M. Lavrauw, J. Sheekey and F. Vanhove, Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces, Elect. J. Comb., 21 (2014), P1.26. |
[19] |
R. Gow and R. Quinlan, Galois theory and linear algebra, Linear Algebra Appl., 430 (2009), 1778-1789.
doi: 10.1016/j.laa.2008.06.030. |
[20] |
R. Gow and R. Quinlan, Galois extensions and subspaces of alternating bilinear forms with special rank properties, Linear Algebra Appl., 430 (2009), 2212-2224.
doi: 10.1016/j.laa.2008.11.021. |
[21] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, in Topics in Finite Fields, 2015, 157-176.
doi: 10.1090/conm/632/12627. |
[22] |
W. M. Kantor, Finite semifields, in Finite Geometries, Groups, and Computation, Walter de Gruyter GmbH & Co. KG, Berlin, 2006, 103-114. |
[23] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[24] |
M. Lavrauw, Scattered spaces in Galois geometry, preprint, arXiv:1512.05251v2 |
[25] |
M. Lavrauw and O. Polverino, Finite semifields, in Current Research Topics in Galois Geometry (eds. J. De Beule and L. Storme), NOVA Academic Publishers, New York, 2011. |
[26] |
M. Lavrauw, J. Sheekey and C. Zanella, On embeddings of minimum dimension of $\PG(n,q)\times \PG(n,q)$, Des. Codes Cryptogr., 74 (2015), 427-440.
doi: 10.1007/s10623-013-9866-8. |
[27] |
H. Liu and T. Honold, A new approach to the main problem of subspace coding, preprint, arXiv:1408.1181 |
[28] |
G. Lunardon, G. Marino, O. Polverino and R. Trombetti, Translation dual of a semifield, J. Combin. Theory Ser. A, 115 (2008), 1321-1332.
doi: 10.1016/j.jcta.2008.02.002. |
[29] |
G. Lunardon and O. Polverino, Blocking sets and derivable partial spreads, J. Algebr. Comb., 14 (2001), 49-56.
doi: 10.1023/A:1011265919847. |
[30] |
G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes, preprint, arXiv:1507.07855v2 |
[31] |
K. Marshall and A-L. Trautmann, Characterizations of MRD and Gabidulin codes, in ALCOMA15, available online at http://user.math.uzh.ch/trautmann/ALCOMA_presentation.pdf |
[32] |
G. Menichetti, On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. Algebra, 47 (1977), 400-410. |
[33] |
K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes, IEEE Trans. Inform. Theory, 60 (2014), 7035-7046.
doi: 10.1109/TIT.2014.2359198. |
[34] |
O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584.
doi: 10.2307/1989849. |
[35] |
K. Otal and F. Özbudak, Some non-Gabidulin MRD codes, in ALCOMA15. |
[36] |
A. Ravagnani, Rank-metric codes and their MacWilliams identities, preprint, arXiv:1410.1333v2 |
[37] |
I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements, Finite Fields Appl., 18 (2012), 1148-1155. |
[38] |
K.-U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory, preprint, arXiv:1410.7184
doi: 10.1007/s10801-015-0595-0. |
[39] |
D. Silva, F. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[40] |
Z. X. Wan, Geometry of Matrices, World Scientific, 1996.
doi: 10.1142/9789812830234. |
show all references
References:
[1] |
J. Ai, T. Honold and H. Liu, The expurgation-augmentation method for constructing good plane subspace codes, preprint, arXiv:1601.01502 |
[2] |
A. A. Albert, Generalized twisted fields, Pacific J. Math., 11 (1961), 1-8. |
[3] |
D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in Proc. ISIT 2013, 509-513. |
[4] |
S. Ball, G. Ebert and M. Lavrauw, A geometric construction of finite semifields, J. Algebra, 311 (2007), 117-129.
doi: 10.1016/j.jalgebra.2006.11.044. |
[5] |
T. Berger, Isometries for rank distance and permutation group of Gabidulin codes, IEEE Trans. Inform. Theory, 49 (2003), 3016-3019.
doi: 10.1109/TIT.2003.819322. |
[6] |
M. Biliotti, V. Jha and N. L. Johnson, The collineation groups of generalized twisted field planes, Geom. Dedicata, 76 (1999), 97-126.
doi: 10.1023/A:1005089016092. |
[7] |
A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $\PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243.
doi: 10.1023/A:1005283806897. |
[8] |
A. Cossidente, G. Marino and F. Pavese, Non-linear maximum rank distance codes, preprint.
doi: 10.1007/s10623-015-0108-0. |
[9] |
J. de la Cruz, M. Kiermaier, A. Wassermann and W. Willems, Algebraic structures of MRD Codes, Adv. Math. Commun., 10 (2016), 499-510.
doi: 10.3934/amc.2016021. |
[10] |
P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.
doi: 10.1016/0097-3165(78)90015-8. |
[11] | |
[12] |
U. Dempwolff, Translation Planes of Small Order, http://www.mathematik.uni-kl.de/~dempw/dempw_Plane.html |
[13] |
J.-G. Dumas, R. Gow, G. McGuire and J. Sheekey, Subspaces of matrices with special rank properties, Linear Algebra Appl., 433 (2010), 191-202.
doi: 10.1016/j.laa.2010.02.015. |
[14] |
E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12. |
[15] |
E. Gabidulin and A. Kshevetskiy, The new construction of rank codes, in Proc. ISIT 2005. |
[16] |
E. M. Gabidulin and N. I. Pilipchuk, Symmetric rank codes, Probl. Inf. Transm., 40 (2004), 103-117.
doi: 10.1023/B:PRIT.0000043925.67309.c6. |
[17] |
M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.
doi: 10.1109/TIT.2010.2048447. |
[18] |
R. Gow, M. Lavrauw, J. Sheekey and F. Vanhove, Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces, Elect. J. Comb., 21 (2014), P1.26. |
[19] |
R. Gow and R. Quinlan, Galois theory and linear algebra, Linear Algebra Appl., 430 (2009), 1778-1789.
doi: 10.1016/j.laa.2008.06.030. |
[20] |
R. Gow and R. Quinlan, Galois extensions and subspaces of alternating bilinear forms with special rank properties, Linear Algebra Appl., 430 (2009), 2212-2224.
doi: 10.1016/j.laa.2008.11.021. |
[21] |
T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, in Topics in Finite Fields, 2015, 157-176.
doi: 10.1090/conm/632/12627. |
[22] |
W. M. Kantor, Finite semifields, in Finite Geometries, Groups, and Computation, Walter de Gruyter GmbH & Co. KG, Berlin, 2006, 103-114. |
[23] |
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.
doi: 10.1109/TIT.2008.926449. |
[24] |
M. Lavrauw, Scattered spaces in Galois geometry, preprint, arXiv:1512.05251v2 |
[25] |
M. Lavrauw and O. Polverino, Finite semifields, in Current Research Topics in Galois Geometry (eds. J. De Beule and L. Storme), NOVA Academic Publishers, New York, 2011. |
[26] |
M. Lavrauw, J. Sheekey and C. Zanella, On embeddings of minimum dimension of $\PG(n,q)\times \PG(n,q)$, Des. Codes Cryptogr., 74 (2015), 427-440.
doi: 10.1007/s10623-013-9866-8. |
[27] |
H. Liu and T. Honold, A new approach to the main problem of subspace coding, preprint, arXiv:1408.1181 |
[28] |
G. Lunardon, G. Marino, O. Polverino and R. Trombetti, Translation dual of a semifield, J. Combin. Theory Ser. A, 115 (2008), 1321-1332.
doi: 10.1016/j.jcta.2008.02.002. |
[29] |
G. Lunardon and O. Polverino, Blocking sets and derivable partial spreads, J. Algebr. Comb., 14 (2001), 49-56.
doi: 10.1023/A:1011265919847. |
[30] |
G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes, preprint, arXiv:1507.07855v2 |
[31] |
K. Marshall and A-L. Trautmann, Characterizations of MRD and Gabidulin codes, in ALCOMA15, available online at http://user.math.uzh.ch/trautmann/ALCOMA_presentation.pdf |
[32] |
G. Menichetti, On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. Algebra, 47 (1977), 400-410. |
[33] |
K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes, IEEE Trans. Inform. Theory, 60 (2014), 7035-7046.
doi: 10.1109/TIT.2014.2359198. |
[34] |
O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584.
doi: 10.2307/1989849. |
[35] |
K. Otal and F. Özbudak, Some non-Gabidulin MRD codes, in ALCOMA15. |
[36] |
A. Ravagnani, Rank-metric codes and their MacWilliams identities, preprint, arXiv:1410.1333v2 |
[37] |
I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements, Finite Fields Appl., 18 (2012), 1148-1155. |
[38] |
K.-U. Schmidt, Symmetric bilinear forms over finite fields with applications to coding theory, preprint, arXiv:1410.7184
doi: 10.1007/s10801-015-0595-0. |
[39] |
D. Silva, F. R. Kschischang and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.
doi: 10.1109/TIT.2008.928291. |
[40] |
Z. X. Wan, Geometry of Matrices, World Scientific, 1996.
doi: 10.1142/9789812830234. |
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