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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]

G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, 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]

G. Berhuy and F. Oggier, An Introduction to Central Simple Simple Algebras and their Applications to Wireless Communications, AMS., Providence, 2013.  Google Scholar

[3]

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

[4]

B. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun., 48 (2000), 2041-2052. Google Scholar

[5]

B. Hughes, Differential space-time modulation, IEEE Trans. Inform. Theory, 46 (2000), 2567-2078. Google Scholar

[6]

M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS, 1998.  Google Scholar

[7]

F. Oggier, Cyclic algebras for noncoherent differential space-time coding, IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.  Google Scholar

[8]

F. Oggier, A survey of algebraic unitary codes, in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.  Google Scholar

[9]

F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic Division Algebras: A Tool for Space-Time Coding, Now Publishers Inc., Hanover, USA, 2007. Google Scholar

[10]

F. Oggier and L. Lequeu, Families of unitary matrices achieving full diversity, in International Symposium on Information Theory, 2005, 1173-1177. Google Scholar

[11]

S. Pumpluen and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.  Google Scholar

[12]

B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.  Google Scholar

[13]

B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

[14]

A. Shokrollahi, B. Hassibi, B. M. Hochwald and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.  Google Scholar

[15]

R. Slessor, Performance of Codes Based on Crossed Product Algebras, Ph.D thesis, School of Mathematics, University of Southampton, 2011. Google Scholar

show all references

References:
[1]

G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, 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]

G. Berhuy and F. Oggier, An Introduction to Central Simple Simple Algebras and their Applications to Wireless Communications, AMS., Providence, 2013.  Google Scholar

[3]

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

[4]

B. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun., 48 (2000), 2041-2052. Google Scholar

[5]

B. Hughes, Differential space-time modulation, IEEE Trans. Inform. Theory, 46 (2000), 2567-2078. Google Scholar

[6]

M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS, 1998.  Google Scholar

[7]

F. Oggier, Cyclic algebras for noncoherent differential space-time coding, IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.  Google Scholar

[8]

F. Oggier, A survey of algebraic unitary codes, in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.  Google Scholar

[9]

F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic Division Algebras: A Tool for Space-Time Coding, Now Publishers Inc., Hanover, USA, 2007. Google Scholar

[10]

F. Oggier and L. Lequeu, Families of unitary matrices achieving full diversity, in International Symposium on Information Theory, 2005, 1173-1177. Google Scholar

[11]

S. Pumpluen and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.  Google Scholar

[12]

B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.  Google Scholar

[13]

B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

[14]

A. Shokrollahi, B. Hassibi, B. M. Hochwald and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.  Google Scholar

[15]

R. Slessor, Performance of Codes Based on Crossed Product Algebras, Ph.D thesis, School of Mathematics, University of Southampton, 2011. Google Scholar

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