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.  Google Scholar

[2]

A. A. Albert, Quadratic forms permitting composition,, Ann. Math., 43 (1942), 161.  doi: 10.2307/1968887.  Google Scholar

[3]

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

[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.   Google Scholar

[5]

J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding,, in, (2003).   Google Scholar

[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.  Google Scholar

[7]

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

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[12]

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

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[16]

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[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.  Google Scholar

[18]

H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels,, in, (2007).   Google Scholar

[19]

J. S. Milne, "Algebraic Number Theory (v3.02),'', (2009), (2009).   Google Scholar

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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).   Google Scholar

[21]

F. Oggier, On the optimality of the golden code,, in, (2006), 468.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

R. D. Schafer, "An Introduction to Nonassociative Algebras,'', Dover Publications Inc., (1995).   Google Scholar

[26]

B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern,, Arch. Math. (Basel), 52 (1989), 245.   Google Scholar

[27]

S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'', with an appendix by A. Pethö, (2003).   Google Scholar

[28]

B. A. Sethuraman, Division algebras and wireless communication,, Notices Amer. Math. Soc., 57 (2010), 1432.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[33]

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

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. A. Albert, Quadratic forms permitting composition,, Ann. Math., 43 (1942), 161.  doi: 10.2307/1968887.  Google Scholar

[3]

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

[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.   Google Scholar

[5]

J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding,, in, (2003).   Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

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

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

H. Lu, Optimal code constructions for SIMO-OFDM frequency selective fading channels,, in, (2007).   Google Scholar

[19]

J. S. Milne, "Algebraic Number Theory (v3.02),'', (2009), (2009).   Google Scholar

[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).   Google Scholar

[21]

F. Oggier, On the optimality of the golden code,, in, (2006), 468.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

R. D. Schafer, "An Introduction to Nonassociative Algebras,'', Dover Publications Inc., (1995).   Google Scholar

[26]

B. Schmal, Diskriminanten, $\mathbbZ$-Ganzheitsbasen und relative Ganzheitsbasen bei multiquadratischen Zahlkörpern,, Arch. Math. (Basel), 52 (1989), 245.   Google Scholar

[27]

S. Schmitt and H. G. Zimmer, "Elliptic Curves. A Computational Approach,'', with an appendix by A. Pethö, (2003).   Google Scholar

[28]

B. A. Sethuraman, Division algebras and wireless communication,, Notices Amer. Math. Soc., 57 (2010), 1432.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[33]

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

[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.  Google Scholar

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