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

1. 

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

Received  July 2013 Revised  March 2014 Published  August 2014

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.
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.   Google Scholar

[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.  doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[3]

G. Berhuy and F. Oggier, On the existence of perfect space-time codes,, IEEE Trans. Inf. Theory, 55 (2009), 2078.  doi: 10.1109/TIT.2009.2016033.  Google Scholar

[4]

G. Berhuy and F. Oggier, Introduction to Central Simple Algebras and their Applications to Wireless Communication,, AMS, (2013).   Google Scholar

[5]

A. Deajim and D. Grant, Space-time codes and nonassociative division algebras over elliptic curves,, Contemp. Math., 463 (2008), 29.  doi: 10.1090/conm/463/09044.  Google Scholar

[6]

L. E. Dickson, Linear Algebras with associativity not assumed,, Duke Math. J., 1 (1935), 113.  doi: 10.1215/S0012-7094-35-00112-0.  Google Scholar

[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, (2005), 722.   Google Scholar

[8]

C. Hollanti, J. Lahtonen, K. Rauto and R. Vehkalahti, Optimal lattices for MIMO codes from division algebras,, IEEE International Symposium on Information Theory, (2006), 783.   Google Scholar

[9]

T. Y. Lam, Quadratic Forms over Fields,, AMS, (2005).   Google Scholar

[10]

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

[11]

F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes,, IEEE Trans. Inf. Theory, 32 (2006), 3885.  doi: 10.1109/TIT.2006.880010.  Google Scholar

[12]

S. Pumplün, Tensor products of central simple algebras and fast-decodable space-time block codes,, preprint, ().   Google Scholar

[13]

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

[14]

S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras,, Adv. Math. Commun., 5 (2011), 609.  doi: 10.3934/amc.2011.5.449.  Google Scholar

[15]

R. D. Schafer, An introduction to nonassociative algebras,, Dover Publ. Inc., (1995).   Google Scholar

[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.  doi: 10.1109/TIT.2003.817831.  Google Scholar

[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.  doi: 10.1109/TIT.2013.2292513.  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, (2012), 192.   Google Scholar

[20]

R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras,, IEEE Trans. Inf. 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]

A. A. Albert, On the power-associativity of rings,, Summa Brazil. Math., 2 (1948), 21.   Google Scholar

[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.  doi: 10.1007/978-3-540-77224-8_13.  Google Scholar

[3]

G. Berhuy and F. Oggier, On the existence of perfect space-time codes,, IEEE Trans. Inf. Theory, 55 (2009), 2078.  doi: 10.1109/TIT.2009.2016033.  Google Scholar

[4]

G. Berhuy and F. Oggier, Introduction to Central Simple Algebras and their Applications to Wireless Communication,, AMS, (2013).   Google Scholar

[5]

A. Deajim and D. Grant, Space-time codes and nonassociative division algebras over elliptic curves,, Contemp. Math., 463 (2008), 29.  doi: 10.1090/conm/463/09044.  Google Scholar

[6]

L. E. Dickson, Linear Algebras with associativity not assumed,, Duke Math. J., 1 (1935), 113.  doi: 10.1215/S0012-7094-35-00112-0.  Google Scholar

[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, (2005), 722.   Google Scholar

[8]

C. Hollanti, J. Lahtonen, K. Rauto and R. Vehkalahti, Optimal lattices for MIMO codes from division algebras,, IEEE International Symposium on Information Theory, (2006), 783.   Google Scholar

[9]

T. Y. Lam, Quadratic Forms over Fields,, AMS, (2005).   Google Scholar

[10]

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

[11]

F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes,, IEEE Trans. Inf. Theory, 32 (2006), 3885.  doi: 10.1109/TIT.2006.880010.  Google Scholar

[12]

S. Pumplün, Tensor products of central simple algebras and fast-decodable space-time block codes,, preprint, ().   Google Scholar

[13]

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

[14]

S. Pumplün and T. Unger, Space-time block codes from nonassociative division algebras,, Adv. Math. Commun., 5 (2011), 609.  doi: 10.3934/amc.2011.5.449.  Google Scholar

[15]

R. D. Schafer, An introduction to nonassociative algebras,, Dover Publ. Inc., (1995).   Google Scholar

[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.  doi: 10.1109/TIT.2003.817831.  Google Scholar

[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.  doi: 10.1109/TIT.2013.2292513.  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, (2012), 192.   Google Scholar

[20]

R. Vehkalahti, C. Hollanti and F. Oggier, Fast-decodable asymmetric space-time codes from division algebras,, IEEE Trans. Inf. 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]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[4]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[5]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[6]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[7]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[8]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[9]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[10]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[11]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[12]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[13]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[14]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[15]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[16]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[17]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[19]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[20]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]