Quotients of orders in cyclic algebras and space-time codes

Frédérique Oggier; B. A. Sethuraman

doi:10.3934/amc.2013.7.441

Article Contents

2013, Volume 7, Issue 4: 441-461. Doi: 10.3934/amc.2013.7.441

This issue Previous Article On the dual of (non)-weakly regular bent functions and self-dual bent functions Next Article ($\sigma,\delta$)-codes

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
2.
Department of Mathematics, California State University Northridge, Northridge, CA 91330

Received: October 2012

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 11S45; Secondary: 11T71, 94B40.

Citation:

Related Papers

Cited by

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.