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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

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, ().   Google Scholar

[2]

A. A. Albert, Generalized twisted fields, Pacific J. Math., 11 (1961), 1-8.  Google Scholar

[3]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in Proc. ISIT 2013, 509-513. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Cossidente, G. Marino and F. Pavese, Non-linear maximum rank distance codes,, preprint., ().  doi: 10.1007/s10623-015-0108-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

U. Dempwolff, Translation Planes of Small Order,, , ().   Google Scholar

[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.  Google Scholar

[14]

E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12.  Google Scholar

[15]

E. Gabidulin and A. Kshevetskiy, The new construction of rank codes, in Proc. ISIT 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. M. Kantor, Finite semifields, in Finite Geometries, Groups, and Computation, Walter de Gruyter GmbH & Co. KG, Berlin, 2006, 103-114.  Google Scholar

[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.  Google Scholar

[24]

M. Lavrauw, Scattered spaces in Galois geometry,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Liu and T. Honold, A new approach to the main problem of subspace coding,, preprint, ().   Google Scholar

[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.  Google Scholar

[29]

G. Lunardon and O. Polverino, Blocking sets and derivable partial spreads, J. Algebr. Comb., 14 (2001), 49-56. doi: 10.1023/A:1011265919847.  Google Scholar

[30]

G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes,, preprint, ().   Google Scholar

[31]

K. Marshall and A-L. Trautmann, Characterizations of MRD and Gabidulin codes,, in ALCOMA15, ().   Google Scholar

[32]

G. Menichetti, On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. Algebra, 47 (1977), 400-410.  Google Scholar

[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.  Google Scholar

[34]

O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849.  Google Scholar

[35]

K. Otal and F. Özbudak, Some non-Gabidulin MRD codes,, in ALCOMA15., ().   Google Scholar

[36]

A. Ravagnani, Rank-metric codes and their MacWilliams identities,, preprint, ().   Google Scholar

[37]

I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements, Finite Fields Appl., 18 (2012), 1148-1155.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

show all references

References:
[1]

J. Ai, T. Honold and H. Liu, The expurgation-augmentation method for constructing good plane subspace codes,, preprint, ().   Google Scholar

[2]

A. A. Albert, Generalized twisted fields, Pacific J. Math., 11 (1961), 1-8.  Google Scholar

[3]

D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero, in Proc. ISIT 2013, 509-513. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Cossidente, G. Marino and F. Pavese, Non-linear maximum rank distance codes,, preprint., ().  doi: 10.1007/s10623-015-0108-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

U. Dempwolff, Translation Planes of Small Order,, , ().   Google Scholar

[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.  Google Scholar

[14]

E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Transm., 21 (1985), 1-12.  Google Scholar

[15]

E. Gabidulin and A. Kshevetskiy, The new construction of rank codes, in Proc. ISIT 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. M. Kantor, Finite semifields, in Finite Geometries, Groups, and Computation, Walter de Gruyter GmbH & Co. KG, Berlin, 2006, 103-114.  Google Scholar

[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.  Google Scholar

[24]

M. Lavrauw, Scattered spaces in Galois geometry,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Liu and T. Honold, A new approach to the main problem of subspace coding,, preprint, ().   Google Scholar

[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.  Google Scholar

[29]

G. Lunardon and O. Polverino, Blocking sets and derivable partial spreads, J. Algebr. Comb., 14 (2001), 49-56. doi: 10.1023/A:1011265919847.  Google Scholar

[30]

G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted Gabidulin codes,, preprint, ().   Google Scholar

[31]

K. Marshall and A-L. Trautmann, Characterizations of MRD and Gabidulin codes,, in ALCOMA15, ().   Google Scholar

[32]

G. Menichetti, On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field, J. Algebra, 47 (1977), 400-410.  Google Scholar

[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.  Google Scholar

[34]

O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc., 35 (1933), 559-584. doi: 10.2307/1989849.  Google Scholar

[35]

K. Otal and F. Özbudak, Some non-Gabidulin MRD codes,, in ALCOMA15., ().   Google Scholar

[36]

A. Ravagnani, Rank-metric codes and their MacWilliams identities,, preprint, ().   Google Scholar

[37]

I. F. Rúa, E. F. Combarro and J. Ranilla, Determination of division algebras with 243 elements, Finite Fields Appl., 18 (2012), 1148-1155.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

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

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