Quotients of orders in cyclic algebras and space-time codes

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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

Abstract
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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.
[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.
[3]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999.
[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.
[5]	G. J. Janusz, Algebraic Number Fields, Second edition, Amer. Math. Soc., 1996.
[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.
[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.
[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.
[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.
[10]	I. Reiner, Maximal Orders, Academic Press, 1975.
[11]	L. H. Rowen, Ring Theory, Academic Press, 1991.
[12]	O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc., 1950.
[13]	B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.
[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.
[15]	A. Wadsworth, Valuation theory on finite dimensional division algebras, Fields Institute Commu., 32 (2002), 385-449.
[16]	L. C. Washington, Introduction to Ceyclotomic Fields, Springer, 1982. doi: 10.1007/978-1-4684-0133-2.
[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.

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.
[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.
[3]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999.
[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.
[5]	G. J. Janusz, Algebraic Number Fields, Second edition, Amer. Math. Soc., 1996.
[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.
[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.
[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.
[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.
[10]	I. Reiner, Maximal Orders, Academic Press, 1975.
[11]	L. H. Rowen, Ring Theory, Academic Press, 1991.
[12]	O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc., 1950.
[13]	B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.
[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.
[15]	A. Wadsworth, Valuation theory on finite dimensional division algebras, Fields Institute Commu., 32 (2002), 385-449.
[16]	L. C. Washington, Introduction to Ceyclotomic Fields, Springer, 1982. doi: 10.1007/978-1-4684-0133-2.
[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.

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