How to obtain division algebras used for fast-decodable space-time block codes

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August 2014, 8(3): 323-342. doi: 10.3934/amc.2014.8.323

How to obtain division algebras used for fast-decodable space-time block codes

Susanne Pumplün ^1,

1.	School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD

Received July 2013 Revised March 2014 Published August 2014

Abstract
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We present families of unital algebras obtained through a doubling process from a cyclic central simple algebra $D=(K/F,\sigma,c)$, employing a $K$-automorphism $\tau$ and an element $d\in D^\times$. These algebras appear in the construction of iterated space-time block codes. We give conditions when these iterated algebras are division which can be used to construct fully diverse iterated codes. We also briefly look at algebras (and codes) obtained from variations of this method.

Keywords: iterated code., Space-time block code, asymmetric, fast-decodable, non-associative division algebra.

Mathematics Subject Classification: Primary: 17A35, 94B0.

Citation: 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

References:

[1]	A. A. Albert, On the power-associativity of rings, Summa Brazil. Math., 2 (1948), 21-33.
[2]	G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, in AAECC 2007 (eds. S. Boztaş and H.F. Lu), 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.
[3]	G. Berhuy and F. Oggier, On the existence of perfect space-time codes, IEEE Trans. Inf. Theory, 55 (2009), 2078-2082. doi: 10.1109/TIT.2009.2016033.
[4]	G. Berhuy and F. Oggier, Introduction to Central Simple Algebras and their Applications to Wireless Communication, AMS, 2013.
[5]	A. Deajim and D. Grant, Space-time codes and nonassociative division algebras over elliptic curves, Contemp. Math., 463 (2008), 29-44. doi: 10.1090/conm/463/09044.
[6]	L. E. Dickson, Linear Algebras with associativity not assumed, Duke Math. J., 1 (1935), 113-125. doi: 10.1215/S0012-7094-35-00112-0.
[7]	P. Elia, A. Sethuraman and P. V. Kumar, Perfect space-time codes with minimum and non-minimum delay for any number of antennas, in International Conference on Wireless Networks, Communications and Mobile Computing, IEEE, 2005, 722-727.
[8]	C. Hollanti, J. Lahtonen, K. Rauto and R. Vehkalahti, Optimal lattices for MIMO codes from division algebras, IEEE International Symposium on Information Theory, Seattle, 2006, 783-787.
[9]	T. Y. Lam, Quadratic Forms over Fields, AMS, 2005.
[10]	N. Markin and F. Oggier, Iterated space-time code constructions from cyclic algebras, IEEE Trans. Inf. Theory, 59 (2013), 5966-5979. doi: 10.1109/TIT.2013.2266397.
[11]	F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes, IEEE Trans. Inf. Theory, 32 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.
[12]	S. Pumplün, Tensor products of central simple algebras and fast-decodable space-time block codes,, preprint, ().
[13]	S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.
[14]	S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 609-629. doi: 10.3934/amc.2011.5.449.
[15]	R. D. Schafer, An introduction to nonassociative algebras, Dover Publ. Inc., New York, 1995.
[16]	B. A. Sethuraman, B. S. Rajan and V. Sashidhar, 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.
[17]	K. P. Srinath and B. S. Rajan, Fast decodable MIDO codes with large coding gain, in 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), 2013, 2910-2914. doi: 10.1109/TIT.2013.2292513.
[18]	A. Steele, Nonassociative cyclic algebras, Israel J. Math., 200 (2014), 361-387. doi: 10.1007/s11856-014-0021-7.
[19]	A. Steele, S. Pumplün and F. Oggier, MIDO space-time codes from associative and non-associative cyclic algebras, Information Theory Workshop, IEEE, 2012, 192-196.
[20]	R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras, IEEE Trans. Inf. Theory, 58 (2012), 2362-2385. doi: 10.1109/TIT.2011.2176310.
[21]	W. C. Waterhouse, Nonassociative quaternion algebras, Algebras Groups Geom., 4 (1987), 365-378.

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References:

[1]	A. A. Albert, On the power-associativity of rings, Summa Brazil. Math., 2 (1948), 21-33.
[2]	G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, in AAECC 2007 (eds. S. Boztaş and H.F. Lu), 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.
[3]	G. Berhuy and F. Oggier, On the existence of perfect space-time codes, IEEE Trans. Inf. Theory, 55 (2009), 2078-2082. doi: 10.1109/TIT.2009.2016033.
[4]	G. Berhuy and F. Oggier, Introduction to Central Simple Algebras and their Applications to Wireless Communication, AMS, 2013.
[5]	A. Deajim and D. Grant, Space-time codes and nonassociative division algebras over elliptic curves, Contemp. Math., 463 (2008), 29-44. doi: 10.1090/conm/463/09044.
[6]	L. E. Dickson, Linear Algebras with associativity not assumed, Duke Math. J., 1 (1935), 113-125. doi: 10.1215/S0012-7094-35-00112-0.
[7]	P. Elia, A. Sethuraman and P. V. Kumar, Perfect space-time codes with minimum and non-minimum delay for any number of antennas, in International Conference on Wireless Networks, Communications and Mobile Computing, IEEE, 2005, 722-727.
[8]	C. Hollanti, J. Lahtonen, K. Rauto and R. Vehkalahti, Optimal lattices for MIMO codes from division algebras, IEEE International Symposium on Information Theory, Seattle, 2006, 783-787.
[9]	T. Y. Lam, Quadratic Forms over Fields, AMS, 2005.
[10]	N. Markin and F. Oggier, Iterated space-time code constructions from cyclic algebras, IEEE Trans. Inf. Theory, 59 (2013), 5966-5979. doi: 10.1109/TIT.2013.2266397.
[11]	F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes, IEEE Trans. Inf. Theory, 32 (2006), 3885-3902. doi: 10.1109/TIT.2006.880010.
[12]	S. Pumplün, Tensor products of central simple algebras and fast-decodable space-time block codes,, preprint, ().
[13]	S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.
[14]	S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 609-629. doi: 10.3934/amc.2011.5.449.
[15]	R. D. Schafer, An introduction to nonassociative algebras, Dover Publ. Inc., New York, 1995.
[16]	B. A. Sethuraman, B. S. Rajan and V. Sashidhar, 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.
[17]	K. P. Srinath and B. S. Rajan, Fast decodable MIDO codes with large coding gain, in 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), 2013, 2910-2914. doi: 10.1109/TIT.2013.2292513.
[18]	A. Steele, Nonassociative cyclic algebras, Israel J. Math., 200 (2014), 361-387. doi: 10.1007/s11856-014-0021-7.
[19]	A. Steele, S. Pumplün and F. Oggier, MIDO space-time codes from associative and non-associative cyclic algebras, Information Theory Workshop, IEEE, 2012, 192-196.
[20]	R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras, IEEE Trans. Inf. Theory, 58 (2012), 2362-2385. doi: 10.1109/TIT.2011.2176310.
[21]	W. C. Waterhouse, Nonassociative quaternion algebras, Algebras Groups Geom., 4 (1987), 365-378.

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