A new family of linear maximum rank distance codes

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August 2016, 10(3): 475-488. doi: 10.3934/amc.2016019

A new family of linear maximum rank distance codes

John Sheekey ^1,

1.	School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Received April 2015 Revised June 2016 Published August 2016

Abstract
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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.

Keywords: linearized polynomials., Rank metric codes.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019

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]	P. Dembowski, Finite Geometries, Springer, 1968.
[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]	P. Dembowski, Finite Geometries, Springer, 1968.
[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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