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
References:
[1]

J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels, IEEE Trans. Commun., 61 (2013), 3396-3403. doi: 10.1109/TCOMM.2013.061913.120278.  Google Scholar

[2]

D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[3]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999. Google Scholar

[4]

B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J. Algebra, 128 (1990), 126-179. doi: 10.1016/0021-8693(90)90047-R.  Google Scholar

[5]

G. J. Janusz, Algebraic Number Fields, Second edition, Amer. Math. Soc., 1996.  Google Scholar

[6]

L. Luzzi, G. R. B. Othman, J. C. Belfiore and E. Viterbo, Golden space-time block-coded modulation, IEEE Trans. Inf. Theory, 55 (2009), 584-597. doi: 10.1109/TIT.2008.2009846.  Google Scholar

[7]

G. Nebe, E. M. Rains and N. J. A. Sloane, Codes and Invariant Theory, Math. Nachrichten, 274 (2004), 104-116. doi: 10.1002/mana.200310204.  Google Scholar

[8]

F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space time block codes, IEEE Trans. Inf. Theory, 52 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.  Google Scholar

[9]

F. Oggier, P. Solé and J.-C. Belfiore, Codes over matrix rings for space-time coded modulations, IEEE Trans. Inf. Theory, 58 (2012), 734-746. doi: 10.1109/TIT.2011.2173732.  Google Scholar

[10]

I. Reiner, Maximal Orders, Academic Press, 1975.  Google Scholar

[11]

L. H. Rowen, Ring Theory, Academic Press, 1991.  Google Scholar

[12]

O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc., 1950.  Google Scholar

[13]

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

[14]

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

[15]

A. Wadsworth, Valuation theory on finite dimensional division algebras, Fields Institute Commu., 32 (2002), 385-449.  Google Scholar

[16]

L. C. Washington, Introduction to Ceyclotomic Fields, Springer, 1982. doi: 10.1007/978-1-4684-0133-2.  Google Scholar

[17]

A. D. Wyner, The wire-tap channel, Bell Syst. Tech. J., 54 (1975), 1355-1387. doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

show all references

References:
[1]

J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels, IEEE Trans. Commun., 61 (2013), 3396-3403. doi: 10.1109/TCOMM.2013.061913.120278.  Google Scholar

[2]

D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[3]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999. Google Scholar

[4]

B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J. Algebra, 128 (1990), 126-179. doi: 10.1016/0021-8693(90)90047-R.  Google Scholar

[5]

G. J. Janusz, Algebraic Number Fields, Second edition, Amer. Math. Soc., 1996.  Google Scholar

[6]

L. Luzzi, G. R. B. Othman, J. C. Belfiore and E. Viterbo, Golden space-time block-coded modulation, IEEE Trans. Inf. Theory, 55 (2009), 584-597. doi: 10.1109/TIT.2008.2009846.  Google Scholar

[7]

G. Nebe, E. M. Rains and N. J. A. Sloane, Codes and Invariant Theory, Math. Nachrichten, 274 (2004), 104-116. doi: 10.1002/mana.200310204.  Google Scholar

[8]

F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space time block codes, IEEE Trans. Inf. Theory, 52 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.  Google Scholar

[9]

F. Oggier, P. Solé and J.-C. Belfiore, Codes over matrix rings for space-time coded modulations, IEEE Trans. Inf. Theory, 58 (2012), 734-746. doi: 10.1109/TIT.2011.2173732.  Google Scholar

[10]

I. Reiner, Maximal Orders, Academic Press, 1975.  Google Scholar

[11]

L. H. Rowen, Ring Theory, Academic Press, 1991.  Google Scholar

[12]

O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc., 1950.  Google Scholar

[13]

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

[14]

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

[15]

A. Wadsworth, Valuation theory on finite dimensional division algebras, Fields Institute Commu., 32 (2002), 385-449.  Google Scholar

[16]

L. C. Washington, Introduction to Ceyclotomic Fields, Springer, 1982. doi: 10.1007/978-1-4684-0133-2.  Google Scholar

[17]

A. D. Wyner, The wire-tap channel, Bell Syst. Tech. J., 54 (1975), 1355-1387. doi: 10.1002/j.1538-7305.1975.tb02040.x.  Google Scholar

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