Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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 $\mathcal{O}(M^{15})$.
    Mathematics Subject Classification: Primary: 17A35, 94B05.


    \begin{equation} \\ \end{equation}
  • [1]

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


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


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


    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.


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


    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.


    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.


    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.


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


    J.-C. Petit, Sur certains quasi-corps généralisant un type d'anneau-quotient, Séminaire Dubriel. Algèbre et théorie des nombres, 20 (1966/67), 1-18.


    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.


    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.


    S. Pumplün, Tensor products of nonassociative cyclic algebras, Online at arXiv:1504.00194 [math.RA]


    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.


    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.


    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.


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


    A. Steele, Nonassociative cyclic algebras, Israel J. Math., 200 (2014), 361-387.doi: 10.1007/s11856-014-0021-7.


    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.


    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.


    W. C. Waterhouse, Nonassociative quaternion algebras, Algebras Groups Geom., 4 (1987), 365-378.

  • 加载中

Article Metrics

HTML views() PDF downloads(222) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint