• Previous Article
    Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields
  • AMC Home
  • This Issue
  • Next Article
    FORSAKES: A forward-secure authenticated key exchange protocol based on symmetric key-evolving schemes
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. doi: 10.1007/BF02773952. Google Scholar

[2]

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

[3]

N. Jacobson, Finite-dimensional Division Algebras Over Fields,, Springer Verlag, (1996). doi: 10.1007/978-3-642-02429-0. Google Scholar

[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. doi: 10.1109/ISIT.2011.6033869. Google Scholar

[5]

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

[6]

N. Markin and F. Oggier, Iterated space-time code constructions from cyclic algebras,, IEEE Trans. Inf. Theory, 59 (2013), 5966. doi: 10.1109/TIT.2013.2266397. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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. doi: 10.1109/ISIT.2012.6284120. Google Scholar

[16]

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

[17]

R. D. Schafer, An Introduction to Nonassociative Algebras,, Dover Publ., (1995). Google Scholar

[18]

A. Steele, Nonassociative cyclic algebras,, Israel J. Math., 200 (2014), 361. doi: 10.1007/s11856-014-0021-7. Google Scholar

[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. Google Scholar

[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. doi: 10.1109/TIT.2011.2176310. Google Scholar

[21]

W. C. Waterhouse, Nonassociative quaternion algebras,, Algebras Groups Geom., 4 (1987), 365. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

N. Jacobson, Finite-dimensional Division Algebras Over Fields,, Springer Verlag, (1996). doi: 10.1007/978-3-642-02429-0. Google Scholar

[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. doi: 10.1109/ISIT.2011.6033869. Google Scholar

[5]

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

[6]

N. Markin and F. Oggier, Iterated space-time code constructions from cyclic algebras,, IEEE Trans. Inf. Theory, 59 (2013), 5966. doi: 10.1109/TIT.2013.2266397. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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. doi: 10.1109/ISIT.2012.6284120. Google Scholar

[16]

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

[17]

R. D. Schafer, An Introduction to Nonassociative Algebras,, Dover Publ., (1995). Google Scholar

[18]

A. Steele, Nonassociative cyclic algebras,, Israel J. Math., 200 (2014), 361. doi: 10.1007/s11856-014-0021-7. Google Scholar

[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. Google Scholar

[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. doi: 10.1109/TIT.2011.2176310. Google Scholar

[21]

W. C. Waterhouse, Nonassociative quaternion algebras,, Algebras Groups Geom., 4 (1987), 365. Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29

[8]

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

[9]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066

[10]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523

[11]

Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212

[12]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420

[13]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[14]

Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023

[15]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399

[16]

Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599

[17]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713

[18]

Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257

[19]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216

[20]

Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]