A new family of linear maximum rank distance codes

Previous Article
Editorial
AMC Home
This Issue
Next Article
Further results on the classification of MDS codes

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

A new family of linear maximum rank distance codes

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

Received April 2015 Revised June 2016 Published August 2016

Abstract
Related Papers

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. Google Scholar
[3]	D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero,, in Proc. ISIT 2013, (2013), 509. Google Scholar
[4]	S. Ball, G. Ebert and M. Lavrauw, A geometric construction of finite semifields,, J. Algebra, 311 (2007), 117. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[15]	E. Gabidulin and A. Kshevetskiy, The new construction of rank codes,, in Proc. ISIT 2005., (2005). Google Scholar
[16]	E. M. Gabidulin and N. I. Pilipchuk, Symmetric rank codes,, Probl. Inf. Transm., 40 (2004), 103. 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. 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). Google Scholar
[19]	R. Gow and R. Quinlan, Galois theory and linear algebra,, Linear Algebra Appl., 430 (2009), 1778. 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. 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. doi: 10.1090/conm/632/12627. Google Scholar
[22]	W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103. 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. 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)*, (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. 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. 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. 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. 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. doi: 10.1109/TIT.2014.2359198. Google Scholar
[34]	O. Ore, On a special class of polynomials,, Trans. Amer. Math. Soc., 35 (1933), 559. 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. 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. 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. Google Scholar
[3]	D. Augot, P. Loidreau and G. Robert, Rank metric and Gabidulin codes in characteristic zero,, in Proc. ISIT 2013, (2013), 509. Google Scholar
[4]	S. Ball, G. Ebert and M. Lavrauw, A geometric construction of finite semifields,, J. Algebra, 311 (2007), 117. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[15]	E. Gabidulin and A. Kshevetskiy, The new construction of rank codes,, in Proc. ISIT 2005., (2005). Google Scholar
[16]	E. M. Gabidulin and N. I. Pilipchuk, Symmetric rank codes,, Probl. Inf. Transm., 40 (2004), 103. 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. 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). Google Scholar
[19]	R. Gow and R. Quinlan, Galois theory and linear algebra,, Linear Algebra Appl., 430 (2009), 1778. 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. 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. doi: 10.1090/conm/632/12627. Google Scholar
[22]	W. M. Kantor, Finite semifields,, in Finite Geometries, (2006), 103. 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. 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)*, (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. 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. 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. 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. 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. doi: 10.1109/TIT.2014.2359198. Google Scholar
[34]	O. Ore, On a special class of polynomials,, Trans. Amer. Math. Soc., 35 (1933), 559. 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. 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. doi: 10.1109/TIT.2008.928291. Google Scholar
[40]	Z. X. Wan, Geometry of Matrices,, World Scientific, (1996). doi: 10.1142/9789812830234. Google Scholar

[1]	Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042
[2]	Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[3]	Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149
[4]	Susanne Pumplün. Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 2017, 11 (3) : 615-634. doi: 10.3934/amc.2017046
[5]	Kamil Otal, Ferruh Özbudak. Explicit constructions of some non-Gabidulin linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 589-600. doi: 10.3934/amc.2016028
[6]	Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$. Advances in Mathematics of Communications, 2012, 6 (2) : 121-130. doi: 10.3934/amc.2012.6.121
[7]	Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[8]	Elisavet Konstantinou, Aristides Kontogeorgis. Some remarks on the construction of class polynomials. Advances in Mathematics of Communications, 2011, 5 (1) : 109-118. doi: 10.3934/amc.2011.5.109
[9]	Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[10]	Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
[11]	Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[12]	Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014
[13]	Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
[14]	Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.
[15]	Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299
[16]	Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
[17]	Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443
[18]	Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010
[19]	Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667
[20]	Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27.

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences