Quotients of orders in cyclic algebras and space-time codes

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

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
Related Papers

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. 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. 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. doi: 10.1016/0021-8693(90)90047-R. Google Scholar
[5]	G. J. Janusz, Algebraic Number Fields,, Second edition, (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. 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. 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. 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. 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. 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. doi: 10.1109/TIT.2003.817831. Google Scholar
[15]	A. Wadsworth, Valuation theory on finite dimensional division algebras,, Fields Institute Commu., 32 (2002), 385. 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. 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. 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. 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. doi: 10.1016/0021-8693(90)90047-R. Google Scholar
[5]	G. J. Janusz, Algebraic Number Fields,, Second edition, (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. 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. 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. 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. 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. 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. doi: 10.1109/TIT.2003.817831. Google Scholar
[15]	A. Wadsworth, Valuation theory on finite dimensional division algebras,, Fields Institute Commu., 32 (2002), 385. 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. doi: 10.1002/j.1538-7305.1975.tb02040.x. Google Scholar

[1]	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
[2]	Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167
[3]	Susanne Pumplün. How to obtain division algebras used for fast-decodable space-time block codes. Advances in Mathematics of Communications, 2014, 8 (3) : 323-342. doi: 10.3934/amc.2014.8.323
[4]	Susanne Pumplün, Andrew Steele. The nonassociative algebras used to build fast-decodable space-time block codes. Advances in Mathematics of Communications, 2015, 9 (4) : 449-469. doi: 10.3934/amc.2015.9.449
[5]	David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35
[6]	Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[7]	David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[8]	Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255
[9]	Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163
[10]	Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225
[11]	Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[12]	Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021
[13]	Susanne Pumplün. Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 2017, 11 (3) : 615-634. doi: 10.3934/amc.2017046
[14]	Srimathy Srinivasan, Andrew Thangaraj. Codes on planar Tanner graphs. Advances in Mathematics of Communications, 2012, 6 (2) : 131-163. doi: 10.3934/amc.2012.6.131
[15]	M. B. Paterson, D. R. Stinson, R. Wei. Combinatorial batch codes. Advances in Mathematics of Communications, 2009, 3 (1) : 13-27. doi: 10.3934/amc.2009.3.13
[16]	Noam Presman, Simon Litsyn. Recursive descriptions of polar codes. Advances in Mathematics of Communications, 2017, 11 (1) : 1-65. doi: 10.3934/amc.2017001
[17]	Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[18]	Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007
[19]	Olof Heden. A survey of perfect codes. Advances in Mathematics of Communications, 2008, 2 (2) : 223-247. doi: 10.3934/amc.2008.2.223
[20]	JiYoon Jung, Carl Mummert, Elizabeth Niese, Michael Schroeder. On erasure combinatorial batch codes. Advances in Mathematics of Communications, 2018, 12 (1) : 49-65. doi: 10.3934/amc.2018003

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences