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.

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