American Institute of Mathematical Sciences

Advanced
May  2014, 8(2): 167-189. doi: 10.3934/amc.2014.8.167

Algebraic space-time codes based on division algebras with a unitary involution

Grégory Berhuy 1,

1. 

Université Joseph Fourier, Institut Fourier, 100 rue des maths, BP 74, F-38402 Saint Martin d'Hères Cedex, France

Received  February 2013 Published  May 2014

In this paper, we focus on the design of unitary space-time codes achieving full diversity using division algebras, and on the systematic computation of their minimum determinant. We also give examples of such codes with high minimum determinant. Division algebras allow to obtain higher rates than known constructions based on finite groups.
Keywords: space-time codes., algebras with involution, unitary involutions, Skew fields, algebraic coding.
Mathematics Subject Classification: Primary: 12E15; Secondary: 11T71,16W1.
Citation: Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167
References:
[1]

in Proceedings of Applied algebra, algebraic algorithms and error-correcting codes, 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[2]

AMS., Providence, 2013.  Google Scholar

[3]

G. Berhuy and R. Slessor, Optimality of codes based on crossed product algebras,, preprint., ().   Google Scholar

[4]

IEEE Trans. Commun., 48 (2000), 2041-2052. Google Scholar

[5]

IEEE Trans. Inform. Theory, 46 (2000), 2567-2078. Google Scholar

[6]

AMS, 1998.  Google Scholar

[7]

IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.  Google Scholar

[8]

in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.  Google Scholar

[9]

Now Publishers Inc., Hanover, USA, 2007. Google Scholar

[10]

in International Symposium on Information Theory, 2005, 1173-1177. Google Scholar

[11]

Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.  Google Scholar

[12]

Notices AMS, 57 (2010), 1432-1439.  Google Scholar

[13]

IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

[14]

IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.  Google Scholar

[15]

Ph.D thesis, School of Mathematics, University of Southampton, 2011. Google Scholar

show all references

References:
[1]

in Proceedings of Applied algebra, algebraic algorithms and error-correcting codes, 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[2]

AMS., Providence, 2013.  Google Scholar

[3]

G. Berhuy and R. Slessor, Optimality of codes based on crossed product algebras,, preprint., ().   Google Scholar

[4]

IEEE Trans. Commun., 48 (2000), 2041-2052. Google Scholar

[5]

IEEE Trans. Inform. Theory, 46 (2000), 2567-2078. Google Scholar

[6]

AMS, 1998.  Google Scholar

[7]

IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.  Google Scholar

[8]

in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.  Google Scholar

[9]

Now Publishers Inc., Hanover, USA, 2007. Google Scholar

[10]

in International Symposium on Information Theory, 2005, 1173-1177. Google Scholar

[11]

Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.  Google Scholar

[12]

Notices AMS, 57 (2010), 1432-1439.  Google Scholar

[13]

IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

[14]

IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.  Google Scholar

[15]

Ph.D thesis, School of Mathematics, University of Southampton, 2011. Google Scholar

[1]

Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032
[2]

Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449
[3]

Frédérique Oggier, B. A. Sethuraman. Quotients of orders in cyclic algebras and space-time codes. Advances in Mathematics of Communications, 2013, 7 (4) : 441-461. doi: 10.3934/amc.2013.7.441
[4]

David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35
[5]

Susanne Pumplün, Andrew Steele. The nonassociative algebras used to build fast-decodable space-time block codes. Advances in Mathematics of Communications, 2015, 9 (4) : 449-469. doi: 10.3934/amc.2015.9.449
[6]

Susanne Pumplün. How to obtain division algebras used for fast-decodable space-time block codes. Advances in Mathematics of Communications, 2014, 8 (3) : 323-342. doi: 10.3934/amc.2014.8.323
[7]

David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[8]

Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255
[9]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[10]

Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212
[11]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[12]

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
[13]

Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022
[14]

Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323
[15]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[16]

Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599
[17]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[18]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[19]

Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257
[20]

Vincent Astier, Thomas Unger. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements, 2018, 25: 16-26. doi: 10.3934/era.2018.25.003

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences