American Institute of Mathematical Sciences

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November  2013, 7(4): 441-461. doi: 10.3934/amc.2013.7.441

Quotients of orders in cyclic algebras and space-time codes

Frédérique Oggier 1, and  B. A. Sethuraman 2,

1. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
2. 

Department of Mathematics, California State University Northridge, Northridge, CA 91330, United States

Received  October 2012 Published  October 2013

Let $F$ be a number field with ring of integers $\boldsymbol{O}_F$ and $D$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\boldsymbol{O}_F$-order $\Lambda$ in $D$ by two-sided ideals. We reduce the problem to studying the ideal structure of $\Lambda/q^s\Lambda$, where $q$ is a prime ideal in $\boldsymbol{O}_F$, $s\geq 1$. We study the case where $q$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.
Keywords: orders., division algebras, Coset codes, space-time codes, wiretap codes.
Mathematics Subject Classification: Primary: 11S45; Secondary: 11T71, 94B4.
Citation: 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
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show all references

References:
[1]

IEEE Trans. Commun., 61 (2013), 3396-3403. doi: 10.1109/TCOMM.2013.061913.120278.  Google Scholar

[2]

J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[3]

Cambridge University Press, 1999. Google Scholar

[4]

J. Algebra, 128 (1990), 126-179. doi: 10.1016/0021-8693(90)90047-R.  Google Scholar

[5]

Second edition, Amer. Math. Soc., 1996.  Google Scholar

[6]

IEEE Trans. Inf. Theory, 55 (2009), 584-597. doi: 10.1109/TIT.2008.2009846.  Google Scholar

[7]

Math. Nachrichten, 274 (2004), 104-116. doi: 10.1002/mana.200310204.  Google Scholar

[8]

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

[9]

IEEE Trans. Inf. Theory, 58 (2012), 734-746. doi: 10.1109/TIT.2011.2173732.  Google Scholar

[10]

Academic Press, 1975.  Google Scholar

[11]

Academic Press, 1991.  Google Scholar

[12]

Amer. Math. Soc., 1950.  Google Scholar

[13]

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

[14]

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

[15]

Fields Institute Commu., 32 (2002), 385-449.  Google Scholar

[16]

Springer, 1982. doi: 10.1007/978-1-4684-0133-2.  Google Scholar

[17]

Bell Syst. Tech. J., 54 (1975), 1355-1387. doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

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