# American Institute of Mathematical Sciences

August  2011, 5(3): 449-471. doi: 10.3934/amc.2011.5.449

## Space-time block codes from nonassociative division algebras

 1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 2 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Received  June 2010 Revised  May 2011 Published  August 2011

Associative division algebras are a rich source of fully diverse space-time block codes (STBCs). In this paper the systematic construction of fully diverse STBCs from nonassociative algebras is discussed. As examples, families of fully diverse $2\times 2$, $2\times 4$ multiblock and $4\times 4$ STBCs are designed, employing nonassociative quaternion division algebras.
Citation: 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
##### References:
 [1] S. M. Alamouti, A simple transmit diversity technique for wireless communications,, IEEE J. Selected Areas Commun., 16 (1998), 1451. doi: 10.1109/49.730453. [2] A. A. Albert, Quadratic forms permitting composition,, Ann. Math., 43 (1942), 161. doi: 10.2307/1968887. [3] A. A. Albert, On the power-associativity of rings,, Summa Brasil. Math., 2 (1948), 21. [4] S. C. Althoen, K. D. Hansen and L. D. Kugler, C-associative algebras of dimension $4$ over R,, Algebras Groups Geom., 3 (1986), 329. [5] J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding,, in, (2003). [6] J.-C. Belfiore, G. Rekaya and E. Viterbo, The golden code: a $2 \times 2$ full-rate space-time code with nonvanishing determinants,, IEEE Trans. Inform. Theory, 51 (2005), 1432. doi: 10.1109/TIT.2005.844069. [7] G. Berhuy and F. Oggier, "Introduction to Central Simple Algebras and their Applications to Wireless Communication,'', AMS Surveys and Monographs, (). [8] G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree $4$,, in, (2007), 90. doi: 10.1007/978-3-540-77224-8_13. [9] G. Berhuy and F. Oggier, On the existence of perfect space-time codes,, IEEE Trans. Inform. Theory, 55 (2009), 2078. doi: 10.1109/TIT.2009.2016033. [10] R. Bott and J. Milnor, On the parallelizability of the spheres,, Bull. Amer. Math. Soc., 64 (1958), 87. doi: 10.1090/S0002-9904-1958-10166-4. [11] E. Darpö, E. Dieterich and M. Herschend, In which dimensions does a division algebra over a given ground field exist?,, Enseign. Math. (2), 51 (2005), 255. [12] L. E. Dickson, Linear algebras with associativity not assumed,, Duke Math. J., 1 (1935), 113. doi: 10.1215/S0012-7094-35-00112-0. [13] P. Elia, B. A. Sethuraman and P. V. Kumar, Perfect space-time codes with minimum and non-minimum delay for any number of antennas,, in, (2005), 722. doi: 10.1109/WIRLES.2005.1549496. [14] H. Hasse, "Number Theory,'' translated from the third (1969) German edition, reprint of the 1980 English edition,, Edited and with a preface by H. G. Zimmer, (2002). [15] C. Hollanti, J. Lahtonen, K. Ranto and R. Vehkalahti, Optimal matrix lattices for MIMO codes from division algebras,, in, (2006), 783. doi: 10.1109/ISIT.2006.261720. [16] C. Jiménez-Gestal and J. M. Pérez-Izquierdo, Ternary derivations of finite-dimensional real division algebras,, Linear Algebra Appl., 428 (2008), 2192. doi: 10.1016/j.laa.2007.11.019. [17] J. Lahtonen, N. Markin and G. McGuire, Construction of multiblock space-time codes from division algebras with roots of unity as nonnorm elements,, IEEE Trans. Inform. Theory, 54 (2008), 5231. doi: 10.1109/TIT.2008.929963. [18] H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels,, in, (2007). [19] J. S. Milne, "Algebraic Number Theory (v3.02),'', (2009), (2009). [20] J. Neukirch, "Algebraic Number Theory,'' translated from the 1992 German edition and with a note by N. Schappacher,, with a foreword by G. Harder, (1999). [21] F. Oggier, On the optimality of the golden code,, in, (2006), 468. [22] F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic division algebras: a tool for space-time coding,, Found. Trends Commun. Inform. Theory, 4 (2007), 1. doi: 10.1561/0100000016. [23] F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes,, IEEE Trans. Inform. Theory, 52 (2006), 3885. doi: 10.1109/TIT.2006.880010. [24] S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings,, Israel J. Math., 155 (2006), 125. doi: 10.1007/BF02773952. [25] R. D. Schafer, "An Introduction to Nonassociative Algebras,'', Dover Publications Inc., (1995). [26] B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern,, Arch. Math. (Basel), 52 (1989), 245. [27] S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'', with an appendix by A. Pethö, (2003). [28] B. A. Sethuraman, Division algebras and wireless communication,, Notices Amer. Math. Soc., 57 (2010), 1432. [29] B. A. Sethuraman, B. Sundar Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras,, IEEE Trans. Inform. Theory, 49 (2003), 2596. doi: 10.1109/TIT.2003.817831. [30] V. Tarokh, H. Jafarkhani and A. R. Calderbank, Space-time block codes from orthogonal designs,, IEEE Trans. Inform. Theory, 45 (1999), 1456. doi: 10.1109/18.771146. [31] V. Tarokh, H. Jafarkhani and A. R. Calderbank, Correction to: "Space-time block codes from orthogonal designs'' [IEEE Trans. Inform. Theory, 45 (1999), 1456-1467],, IEEE Trans. Inform. Theory, 46 (2000). doi: 10.1109/TIT.2000.1282193. [32] T. Unger and N. Markin, Quadratic forms and space-time block codes from generalized quaternion and biquaternion algebras,, IEEE Trans. Inform. Theory, 57 (2011). [33] W. C. Waterhouse, Nonassociative quaternion algebras,, Algebras Groups Geom., 4 (1987), 365. [34] K. Yamamura, The determination of the imaginary abelian number fields with class number one,, Math. Comp., 62 (1994), 899. doi: 10.1090/S0025-5718-1994-1218347-3.

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##### References:
 [1] S. M. Alamouti, A simple transmit diversity technique for wireless communications,, IEEE J. Selected Areas Commun., 16 (1998), 1451. doi: 10.1109/49.730453. [2] A. A. Albert, Quadratic forms permitting composition,, Ann. Math., 43 (1942), 161. doi: 10.2307/1968887. [3] A. A. Albert, On the power-associativity of rings,, Summa Brasil. Math., 2 (1948), 21. [4] S. C. Althoen, K. D. Hansen and L. D. Kugler, C-associative algebras of dimension $4$ over R,, Algebras Groups Geom., 3 (1986), 329. [5] J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding,, in, (2003). [6] J.-C. Belfiore, G. Rekaya and E. Viterbo, The golden code: a $2 \times 2$ full-rate space-time code with nonvanishing determinants,, IEEE Trans. Inform. Theory, 51 (2005), 1432. doi: 10.1109/TIT.2005.844069. [7] G. Berhuy and F. Oggier, "Introduction to Central Simple Algebras and their Applications to Wireless Communication,'', AMS Surveys and Monographs, (). [8] G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree $4$,, in, (2007), 90. doi: 10.1007/978-3-540-77224-8_13. [9] G. Berhuy and F. Oggier, On the existence of perfect space-time codes,, IEEE Trans. Inform. Theory, 55 (2009), 2078. doi: 10.1109/TIT.2009.2016033. [10] R. Bott and J. Milnor, On the parallelizability of the spheres,, Bull. Amer. Math. Soc., 64 (1958), 87. doi: 10.1090/S0002-9904-1958-10166-4. [11] E. Darpö, E. Dieterich and M. Herschend, In which dimensions does a division algebra over a given ground field exist?,, Enseign. Math. (2), 51 (2005), 255. [12] L. E. Dickson, Linear algebras with associativity not assumed,, Duke Math. J., 1 (1935), 113. doi: 10.1215/S0012-7094-35-00112-0. [13] P. Elia, B. A. Sethuraman and P. V. Kumar, Perfect space-time codes with minimum and non-minimum delay for any number of antennas,, in, (2005), 722. doi: 10.1109/WIRLES.2005.1549496. [14] H. Hasse, "Number Theory,'' translated from the third (1969) German edition, reprint of the 1980 English edition,, Edited and with a preface by H. G. Zimmer, (2002). [15] C. Hollanti, J. Lahtonen, K. Ranto and R. Vehkalahti, Optimal matrix lattices for MIMO codes from division algebras,, in, (2006), 783. doi: 10.1109/ISIT.2006.261720. [16] C. Jiménez-Gestal and J. M. Pérez-Izquierdo, Ternary derivations of finite-dimensional real division algebras,, Linear Algebra Appl., 428 (2008), 2192. doi: 10.1016/j.laa.2007.11.019. [17] J. Lahtonen, N. Markin and G. McGuire, Construction of multiblock space-time codes from division algebras with roots of unity as nonnorm elements,, IEEE Trans. Inform. Theory, 54 (2008), 5231. doi: 10.1109/TIT.2008.929963. [18] H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels,, in, (2007). [19] J. S. Milne, "Algebraic Number Theory (v3.02),'', (2009), (2009). [20] J. Neukirch, "Algebraic Number Theory,'' translated from the 1992 German edition and with a note by N. Schappacher,, with a foreword by G. Harder, (1999). [21] F. Oggier, On the optimality of the golden code,, in, (2006), 468. [22] F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic division algebras: a tool for space-time coding,, Found. Trends Commun. Inform. Theory, 4 (2007), 1. doi: 10.1561/0100000016. [23] F. Oggier, G. Rekaya, J.-C. Belfiore and E. Viterbo, Perfect space-time block codes,, IEEE Trans. Inform. Theory, 52 (2006), 3885. doi: 10.1109/TIT.2006.880010. [24] S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings,, Israel J. Math., 155 (2006), 125. doi: 10.1007/BF02773952. [25] R. D. Schafer, "An Introduction to Nonassociative Algebras,'', Dover Publications Inc., (1995). [26] B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern,, Arch. Math. (Basel), 52 (1989), 245. [27] S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'', with an appendix by A. Pethö, (2003). [28] B. A. Sethuraman, Division algebras and wireless communication,, Notices Amer. Math. Soc., 57 (2010), 1432. [29] B. A. Sethuraman, B. Sundar Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras,, IEEE Trans. Inform. Theory, 49 (2003), 2596. doi: 10.1109/TIT.2003.817831. [30] V. Tarokh, H. Jafarkhani and A. R. Calderbank, Space-time block codes from orthogonal designs,, IEEE Trans. Inform. Theory, 45 (1999), 1456. doi: 10.1109/18.771146. [31] V. Tarokh, H. Jafarkhani and A. R. Calderbank, Correction to: "Space-time block codes from orthogonal designs'' [IEEE Trans. Inform. Theory, 45 (1999), 1456-1467],, IEEE Trans. Inform. Theory, 46 (2000). doi: 10.1109/TIT.2000.1282193. [32] T. Unger and N. Markin, Quadratic forms and space-time block codes from generalized quaternion and biquaternion algebras,, IEEE Trans. Inform. Theory, 57 (2011). [33] W. C. Waterhouse, Nonassociative quaternion algebras,, Algebras Groups Geom., 4 (1987), 365. [34] K. Yamamura, The determination of the imaginary abelian number fields with class number one,, Math. Comp., 62 (1994), 899. doi: 10.1090/S0025-5718-1994-1218347-3.
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