August  2011, 5(3): 449-471. doi: 10.3934/amc.2011.5.449

Space-time block codes from nonassociative division algebras

1. 

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

2. 

School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Received  June 2010 Revised  May 2011 Published  August 2011

Associative division algebras are a rich source of fully diverse space-time block codes (STBCs). In this paper the systematic construction of fully diverse STBCs from nonassociative algebras is discussed. As examples, families of fully diverse $2\times 2$, $2\times 4$ multiblock and $4\times 4$ STBCs are designed, employing nonassociative quaternion division algebras.
Citation: 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
References:
[1]

IEEE J. Selected Areas Commun., 16 (1998), 1451-1458. doi: 10.1109/49.730453.  Google Scholar

[2]

Ann. Math., 43 (1942), 161-177. doi: 10.2307/1968887.  Google Scholar

[3]

Summa Brasil. Math., 2 (1948), 21-32.  Google Scholar

[4]

Algebras Groups Geom., 3 (1986), 329-360.  Google Scholar

[5]

in "Proceedings of the Information Theory Workshop,'' IEEE, Paris, (2003). Google Scholar

[6]

IEEE Trans. Inform. Theory, 51 (2005), 1432-1436. doi: 10.1109/TIT.2005.844069.  Google Scholar

[7]

G. Berhuy and F. Oggier, "Introduction to Central Simple Algebras and their Applications to Wireless Communication,'', AMS Surveys and Monographs, ().   Google Scholar

[8]

in "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (2007), 90-99. doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[9]

IEEE Trans. Inform. Theory, 55 (2009), 2078-2082. doi: 10.1109/TIT.2009.2016033.  Google Scholar

[10]

Bull. Amer. Math. Soc., 64 (1958), 87-89. doi: 10.1090/S0002-9904-1958-10166-4.  Google Scholar

[11]

Enseign. Math. (2), 51 (2005), 255-263.  Google Scholar

[12]

Duke Math. J., 1 (1935), 113-125. doi: 10.1215/S0012-7094-35-00112-0.  Google Scholar

[13]

in "2005 International Conference on Wireless Networks, Communications and Mobile Computing,'' (2005), 722-727. doi: 10.1109/WIRLES.2005.1549496.  Google Scholar

[14]

Edited and with a preface by H. G. Zimmer, Springer-Verlag, Berlin, 2002.  Google Scholar

[15]

in "2006 IEEE International Symposium on Information Theory,'' (2006), 783-787. doi: 10.1109/ISIT.2006.261720.  Google Scholar

[16]

Linear Algebra Appl., 428 (2008), 2192-2219. doi: 10.1016/j.laa.2007.11.019.  Google Scholar

[17]

IEEE Trans. Inform. Theory, 54 (2008), 5231-5235. doi: 10.1109/TIT.2008.929963.  Google Scholar

[18]

in "Information Theory for Wireless Networks, 2007 IEEE Information Theory Workshop,'' (2007). Google Scholar

[19]

(2009), available online at www.jmilne.org/math/ Google Scholar

[20]

with a foreword by G. Harder, Springer-Verlag, Berlin, 1999.  Google Scholar

[21]

in "Information Theory Workshop, ITW '06,'' Chengdu, (2006), 468-472. Google Scholar

[22]

Found. Trends Commun. Inform. Theory, 4 (2007), 1-95. doi: 10.1561/0100000016.  Google Scholar

[23]

IEEE Trans. Inform. Theory, 52 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.  Google Scholar

[24]

Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.  Google Scholar

[25]

Dover Publications Inc., New York, 1995.  Google Scholar

[26]

Arch. Math. (Basel), 52 (1989), 245-257.  Google Scholar

[27]

with an appendix by A. Pethö, Walter de Gruyter & Co., Berlin, 2003.  Google Scholar

[28]

Notices Amer. Math. Soc., 57 (2010), 1432-1439.  Google Scholar

[29]

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

[30]

IEEE Trans. Inform. Theory, 45 (1999), 1456-1467. doi: 10.1109/18.771146.  Google Scholar

[31]

IEEE Trans. Inform. Theory, 46 (2000), 314. doi: 10.1109/TIT.2000.1282193.  Google Scholar

[32]

IEEE Trans. Inform. Theory, 57 (2011), to appear. Google Scholar

[33]

Algebras Groups Geom., 4 (1987), 365-378.  Google Scholar

[34]

Math. Comp., 62 (1994), 899-921. doi: 10.1090/S0025-5718-1994-1218347-3.  Google Scholar

show all references

References:
[1]

IEEE J. Selected Areas Commun., 16 (1998), 1451-1458. doi: 10.1109/49.730453.  Google Scholar

[2]

Ann. Math., 43 (1942), 161-177. doi: 10.2307/1968887.  Google Scholar

[3]

Summa Brasil. Math., 2 (1948), 21-32.  Google Scholar

[4]

Algebras Groups Geom., 3 (1986), 329-360.  Google Scholar

[5]

in "Proceedings of the Information Theory Workshop,'' IEEE, Paris, (2003). Google Scholar

[6]

IEEE Trans. Inform. Theory, 51 (2005), 1432-1436. doi: 10.1109/TIT.2005.844069.  Google Scholar

[7]

G. Berhuy and F. Oggier, "Introduction to Central Simple Algebras and their Applications to Wireless Communication,'', AMS Surveys and Monographs, ().   Google Scholar

[8]

in "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (2007), 90-99. doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[9]

IEEE Trans. Inform. Theory, 55 (2009), 2078-2082. doi: 10.1109/TIT.2009.2016033.  Google Scholar

[10]

Bull. Amer. Math. Soc., 64 (1958), 87-89. doi: 10.1090/S0002-9904-1958-10166-4.  Google Scholar

[11]

Enseign. Math. (2), 51 (2005), 255-263.  Google Scholar

[12]

Duke Math. J., 1 (1935), 113-125. doi: 10.1215/S0012-7094-35-00112-0.  Google Scholar

[13]

in "2005 International Conference on Wireless Networks, Communications and Mobile Computing,'' (2005), 722-727. doi: 10.1109/WIRLES.2005.1549496.  Google Scholar

[14]

Edited and with a preface by H. G. Zimmer, Springer-Verlag, Berlin, 2002.  Google Scholar

[15]

in "2006 IEEE International Symposium on Information Theory,'' (2006), 783-787. doi: 10.1109/ISIT.2006.261720.  Google Scholar

[16]

Linear Algebra Appl., 428 (2008), 2192-2219. doi: 10.1016/j.laa.2007.11.019.  Google Scholar

[17]

IEEE Trans. Inform. Theory, 54 (2008), 5231-5235. doi: 10.1109/TIT.2008.929963.  Google Scholar

[18]

in "Information Theory for Wireless Networks, 2007 IEEE Information Theory Workshop,'' (2007). Google Scholar

[19]

(2009), available online at www.jmilne.org/math/ Google Scholar

[20]

with a foreword by G. Harder, Springer-Verlag, Berlin, 1999.  Google Scholar

[21]

in "Information Theory Workshop, ITW '06,'' Chengdu, (2006), 468-472. Google Scholar

[22]

Found. Trends Commun. Inform. Theory, 4 (2007), 1-95. doi: 10.1561/0100000016.  Google Scholar

[23]

IEEE Trans. Inform. Theory, 52 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.  Google Scholar

[24]

Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.  Google Scholar

[25]

Dover Publications Inc., New York, 1995.  Google Scholar

[26]

Arch. Math. (Basel), 52 (1989), 245-257.  Google Scholar

[27]

with an appendix by A. Pethö, Walter de Gruyter & Co., Berlin, 2003.  Google Scholar

[28]

Notices Amer. Math. Soc., 57 (2010), 1432-1439.  Google Scholar

[29]

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

[30]

IEEE Trans. Inform. Theory, 45 (1999), 1456-1467. doi: 10.1109/18.771146.  Google Scholar

[31]

IEEE Trans. Inform. Theory, 46 (2000), 314. doi: 10.1109/TIT.2000.1282193.  Google Scholar

[32]

IEEE Trans. Inform. Theory, 57 (2011), to appear. Google Scholar

[33]

Algebras Groups Geom., 4 (1987), 365-378.  Google Scholar

[34]

Math. Comp., 62 (1994), 899-921. doi: 10.1090/S0025-5718-1994-1218347-3.  Google Scholar

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