# 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), 1451-1458. doi: 10.1109/49.730453. [2] A. A. Albert, Quadratic forms permitting composition, Ann. Math., 43 (1942), 161-177. doi: 10.2307/1968887. [3] A. A. Albert, On the power-associativity of rings, Summa Brasil. Math., 2 (1948), 21-32. [4] S. C. Althoen, K. D. Hansen and L. D. Kugler, C-associative algebras of dimension $4$ over R, Algebras Groups Geom., 3 (1986), 329-360. [5] J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding, in "Proceedings of the Information Theory Workshop,'' IEEE, Paris, (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), 1432-1436. 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 "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (2007), 90-99. 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), 2078-2082. doi: 10.1109/TIT.2009.2016033. [10] R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc., 64 (1958), 87-89. 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), 255-263. [12] L. E. Dickson, Linear algebras with associativity not assumed, Duke Math. J., 1 (1935), 113-125. 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 International Conference on Wireless Networks, Communications and Mobile Computing,'' (2005), 722-727. 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, Springer-Verlag, Berlin, 2002. [15] C. Hollanti, J. Lahtonen, K. Ranto and R. Vehkalahti, Optimal matrix lattices for MIMO codes from division algebras, in "2006 IEEE International Symposium on Information Theory,'' (2006), 783-787. 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), 2192-2219. 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), 5231-5235. doi: 10.1109/TIT.2008.929963. [18] H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels, in "Information Theory for Wireless Networks, 2007 IEEE Information Theory Workshop,'' (2007). [19] J. S. Milne, "Algebraic Number Theory (v3.02),'' (2009), available online at www.jmilne.org/math/ [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, Springer-Verlag, Berlin, 1999. [21] F. Oggier, On the optimality of the golden code, in "Information Theory Workshop, ITW '06,'' Chengdu, (2006), 468-472. [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), 1-95. 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), 3885-3902. doi: 10.1109/TIT.2006.880010. [24] S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952. [25] R. D. Schafer, "An Introduction to Nonassociative Algebras,'' Dover Publications Inc., New York, 1995. [26] B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern, Arch. Math. (Basel), 52 (1989), 245-257. [27] S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'' with an appendix by A. Pethö, Walter de Gruyter & Co., Berlin, 2003. [28] B. A. Sethuraman, Division algebras and wireless communication, Notices Amer. Math. Soc., 57 (2010), 1432-1439. [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), 2596-2616. 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), 1456-1467. 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), 314. 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), to appear. [33] W. C. Waterhouse, Nonassociative quaternion algebras, Algebras Groups Geom., 4 (1987), 365-378. [34] K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp., 62 (1994), 899-921. doi: 10.1090/S0025-5718-1994-1218347-3.

show all references

##### References:
 [1] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Selected Areas Commun., 16 (1998), 1451-1458. doi: 10.1109/49.730453. [2] A. A. Albert, Quadratic forms permitting composition, Ann. Math., 43 (1942), 161-177. doi: 10.2307/1968887. [3] A. A. Albert, On the power-associativity of rings, Summa Brasil. Math., 2 (1948), 21-32. [4] S. C. Althoen, K. D. Hansen and L. D. Kugler, C-associative algebras of dimension $4$ over R, Algebras Groups Geom., 3 (1986), 329-360. [5] J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding, in "Proceedings of the Information Theory Workshop,'' IEEE, Paris, (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), 1432-1436. 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 "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,'' Springer, Berlin, (2007), 90-99. 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), 2078-2082. doi: 10.1109/TIT.2009.2016033. [10] R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc., 64 (1958), 87-89. 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), 255-263. [12] L. E. Dickson, Linear algebras with associativity not assumed, Duke Math. J., 1 (1935), 113-125. 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 International Conference on Wireless Networks, Communications and Mobile Computing,'' (2005), 722-727. 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, Springer-Verlag, Berlin, 2002. [15] C. Hollanti, J. Lahtonen, K. Ranto and R. Vehkalahti, Optimal matrix lattices for MIMO codes from division algebras, in "2006 IEEE International Symposium on Information Theory,'' (2006), 783-787. 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), 2192-2219. 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), 5231-5235. doi: 10.1109/TIT.2008.929963. [18] H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels, in "Information Theory for Wireless Networks, 2007 IEEE Information Theory Workshop,'' (2007). [19] J. S. Milne, "Algebraic Number Theory (v3.02),'' (2009), available online at www.jmilne.org/math/ [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, Springer-Verlag, Berlin, 1999. [21] F. Oggier, On the optimality of the golden code, in "Information Theory Workshop, ITW '06,'' Chengdu, (2006), 468-472. [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), 1-95. 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), 3885-3902. doi: 10.1109/TIT.2006.880010. [24] S. Pumplün and V. Astier, Nonassociative quaternion algebras over rings, Israel J. Math., 155 (2006), 125-147. doi: 10.1007/BF02773952. [25] R. D. Schafer, "An Introduction to Nonassociative Algebras,'' Dover Publications Inc., New York, 1995. [26] B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern, Arch. Math. (Basel), 52 (1989), 245-257. [27] S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'' with an appendix by A. Pethö, Walter de Gruyter & Co., Berlin, 2003. [28] B. A. Sethuraman, Division algebras and wireless communication, Notices Amer. Math. Soc., 57 (2010), 1432-1439. [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), 2596-2616. 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), 1456-1467. 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), 314. 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), to appear. [33] W. C. Waterhouse, Nonassociative quaternion algebras, Algebras Groups Geom., 4 (1987), 365-378. [34] K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp., 62 (1994), 899-921. doi: 10.1090/S0025-5718-1994-1218347-3.
 [1] 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 [2] 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 [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] 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 [5] 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 [6] Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $A_n$-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118 [7] 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 [8] Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79 [9] 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 [10] Susanne Pumplün. Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 2017, 11 (3) : 615-634. doi: 10.3934/amc.2017046 [11] Yuming Zhang. On continuity equations in space-time domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212 [12] 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 [13] Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 [14] Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599 [15] Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713 [16] Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 [17] Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257 [18] Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems and Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 [19] Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001 [20] Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure and Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019

2020 Impact Factor: 0.935