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November  2015, 9(4): 449-469. doi: 10.3934/amc.2015.9.449

The nonassociative algebras used to build fast-decodable space-time block codes

1. 

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

2. 

Flat 203, Wilson Tower, 16 Christian Street, London E1 1AW, United Kingdom

Received  March 2014 Revised  November 2014 Published  November 2015

Let $K/F$ and $K/L$ be two cyclic Galois field extensions and $D=(K/F,\sigma,c)$ a cyclic algebra. Given an invertible element $d\in D$, we present three families of unital nonassociative algebras over $L\cap F$ defined on the direct sum of $n$ copies of $D$. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors. We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-$m$ for $nm$ transmit and $m$ receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most $\mathcal{O}(M^{15})$.
Citation: 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
References:
[1]

V. Astier and S. Pumplün, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.

[2]

C. Brown, PhD Thesis University of Nottingham,, in preparation., (). 

[3]

N. Jacobson, Finite-dimensional Division Algebras Over Fields, Springer Verlag, Berlin-Heidelberg-New York, 1996. doi: 10.1007/978-3-642-02429-0.

[4]

G. R. Jithamitra and B. S. Rajan, Minimizing the complexity of fast-sphere decoding of STBCs, IEEE Int. Symposium on Information Theory Proceedings (ISIT), (2011), 1846-1850. doi: 10.1109/ISIT.2011.6033869.

[5]

M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS Coll. Publications, 44, 1998.

[6]

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.

[7]

L. P. Natarajan and B. S. Rajan, Fast group-decodable STBCs via codes over GF(4), Proc. IEEE Int. Symp. Inform. Theory, Austin, TX, (2010), 1056-1060. doi: 10.1109/ISIT.2010.5513721.

[8]

L. P. Natarajan and B. S. Rajan, Fast-Group-Decodable STBCs via codes over GF(4): Further results, Proceedings of IEEE ICC 2011, (ICC'11), Kyoto, Japan, (2011), 1-6. doi: 10.1109/icc.2011.5962874.

[9]

L. P. Natarajan and B. S. Rajan, written communication,, 2013., (). 

[10]

J.-C. Petit, Sur certains quasi-corps généralisant un type d'anneau-quotient,, Séminaire Dubriel. Algèbre et théorie des nombres, 20 (): 1. 

[11]

S. Pumplün and A. Steele, Fast-decodable MIDO codes from nonassociative algebras, Int. J. of Information and Coding Theory (IJICOT), 3 (2015), 15-38. doi: 10.1504/IJICOT.2015.068695.

[12]

S. Pumplün, How to obtain division algebras used for fast decodable space-time block codes, Adv. Math. Comm., 8 (2014), 323-342. doi: 10.3934/amc.2014.8.323.

[13]

S. Pumplün, Tensor products of nonassociative cyclic algebras,, Online at , (). 

[14]

S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Comm., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.

[15]

K. P. Srinath and B. S. Rajan, DMT-optimal, low ML-complexity STBC-schemes for asymmetric MIMO systems, 2012 IEEE International Symposium on Information Theory Proceedings (ISIT), (2012), 3043-3047. doi: 10.1109/ISIT.2012.6284120.

[16]

K. P. Srinath and B. S. Rajan, Fast-decodable MIDO codes with large coding gain, IEEE Transactions on Information Theory, 60 (2014), 992-1007. doi: 10.1109/TIT.2013.2292513.

[17]

R. D. Schafer, An Introduction to Nonassociative Algebras, Dover Publ., Inc., New York, 1995.

[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 (ITW) 2012 IEEE, (2012), 192-196.

[20]

R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras, IEEE Transactions on Information 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.

show all references

References:
[1]

V. Astier and S. Pumplün, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952.

[2]

C. Brown, PhD Thesis University of Nottingham,, in preparation., (). 

[3]

N. Jacobson, Finite-dimensional Division Algebras Over Fields, Springer Verlag, Berlin-Heidelberg-New York, 1996. doi: 10.1007/978-3-642-02429-0.

[4]

G. R. Jithamitra and B. S. Rajan, Minimizing the complexity of fast-sphere decoding of STBCs, IEEE Int. Symposium on Information Theory Proceedings (ISIT), (2011), 1846-1850. doi: 10.1109/ISIT.2011.6033869.

[5]

M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS Coll. Publications, 44, 1998.

[6]

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.

[7]

L. P. Natarajan and B. S. Rajan, Fast group-decodable STBCs via codes over GF(4), Proc. IEEE Int. Symp. Inform. Theory, Austin, TX, (2010), 1056-1060. doi: 10.1109/ISIT.2010.5513721.

[8]

L. P. Natarajan and B. S. Rajan, Fast-Group-Decodable STBCs via codes over GF(4): Further results, Proceedings of IEEE ICC 2011, (ICC'11), Kyoto, Japan, (2011), 1-6. doi: 10.1109/icc.2011.5962874.

[9]

L. P. Natarajan and B. S. Rajan, written communication,, 2013., (). 

[10]

J.-C. Petit, Sur certains quasi-corps généralisant un type d'anneau-quotient,, Séminaire Dubriel. Algèbre et théorie des nombres, 20 (): 1. 

[11]

S. Pumplün and A. Steele, Fast-decodable MIDO codes from nonassociative algebras, Int. J. of Information and Coding Theory (IJICOT), 3 (2015), 15-38. doi: 10.1504/IJICOT.2015.068695.

[12]

S. Pumplün, How to obtain division algebras used for fast decodable space-time block codes, Adv. Math. Comm., 8 (2014), 323-342. doi: 10.3934/amc.2014.8.323.

[13]

S. Pumplün, Tensor products of nonassociative cyclic algebras,, Online at , (). 

[14]

S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Comm., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.

[15]

K. P. Srinath and B. S. Rajan, DMT-optimal, low ML-complexity STBC-schemes for asymmetric MIMO systems, 2012 IEEE International Symposium on Information Theory Proceedings (ISIT), (2012), 3043-3047. doi: 10.1109/ISIT.2012.6284120.

[16]

K. P. Srinath and B. S. Rajan, Fast-decodable MIDO codes with large coding gain, IEEE Transactions on Information Theory, 60 (2014), 992-1007. doi: 10.1109/TIT.2013.2292513.

[17]

R. D. Schafer, An Introduction to Nonassociative Algebras, Dover Publ., Inc., New York, 1995.

[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 (ITW) 2012 IEEE, (2012), 192-196.

[20]

R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras, IEEE Transactions on Information 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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