Input-state-output representations and constructions of finite support 2D convolutional codes

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November 2010, 4(4): 533-545. doi: 10.3934/amc.2010.4.533

Input-state-output representations and constructions of finite support 2D convolutional codes

Diego Napp ^1, , Carmen Perea ^2, and Raquel Pinto ^1,

1.	Department of Mathematics, University of Aveiro, Campus Universitario de Santiago, 3810-193 Aveiro, Portugal, Portugal
2.	Center of Operation Research, Department of Statistics, Mathematics and Informatics, University Miguel Hernández, Av. Universidad s/n, 0302 Elche, Spain

Received December 2009 Revised June 2010 Published November 2010

Abstract
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Two-dimensional convolutional codes are considered, with codewords having compact support indexed in $\mathbb N$² and taking values in $\mathbb F$ⁿ, where $\mathbb F$ is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented.

Keywords: 2D convolutional codes, input-state-output representations., systems theory.

Mathematics Subject Classification: Primary: 94B10, 93C35; Secondary: 93B2.

Citation: Diego Napp, Carmen Perea, Raquel Pinto. Input-state-output representations and constructions of finite support 2D convolutional codes. Advances in Mathematics of Communications, 2010, 4 (4) : 533-545. doi: 10.3934/amc.2010.4.533

References:

[1]	E. Fornasini and G. Marchesini, Structure and properties of two-dimensional systems, in "Multidimensional Systems, Techniques and Applications'' (ed. S.G. Tzafestas), (1986), 37-88.
[2]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes, IEEE Trans. Inf. Th., 40 (1994), 1068-1082. doi: 10.1109/18.335967.
[3]	H. Gluesing-Luersen, J. Rosenthal and P. A. Weiner, Duality between mutidimensinal convolutional codes and systems, in "Advances in Mathematical Systems Theory, a Volume in Honor of Diedrich Hinrichsen'' (eds. F. Colonius, U. Helmke, F. Wirth and D. Prtzel-Wolters), Birkhuser, Boston, (2000), 135-150.
[4]	J. Justesen and S. Forchhammer, Two dimensional information theory and coding. With applications to graphics data and high-density storage media, Cambridge University Press, Cambridge, (2010), 171.
[5]	T. Kailath, "Linear Systems,'' Prentice Hall, Englewood Cliffs, NJ, 1980.
[6]	B. Kitchens, Multidimensional convolutional codes, SIAM J. Discrete Math., 15 (2002), 367-381. doi: 10.1137/S0895480100378495.
[7]	B. C. Lévy, "2-D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,'' Ph.D thesis, Stanford University, USA, 1981.
[8]	R. G. Lobo, "On Locally Invertible Encoders and Muldimensional Convolutional Codes,'' Ph.D thesis, North Carolina State University, USA, 2006.
[9]	P. Rocha, "Structure and Representation of 2-D Systems,'' Ph.D thesis, Groningen University, Holland, 1990.
[10]	J. Rosenthal, J. M. Schumacher and E. V. York, On behaviors and convolutional codes, IEEE Trans. Inf. Th., 42 (1996), 1881-1891. doi: 10.1109/18.556682.
[11]	J. Rosenthal and R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput., 10 (1999), 15-32. doi: 10.1007/s002000050120.
[12]	J. Rosenthal and E. V. York, BCH convolutional codes, IEEE Trans. Inf. Th., 45 (1999), 1833-1844. doi: 10.1109/18.782104.
[13]	J. Singh and M. L. Singh, A new family of two-dimensional codes for optical CDMA systems, Optik - International Journal for Light and Electron Optics, 120 (2009), 959-962.
[14]	J. Swartz, T. Pavlidis and Y. P. Wang, Information encoding with two-dimensional bar codes, IEEE Computer Society, 25 (1992), 18-28.
[15]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes: an algebraic approach, Multidimensional Systems and Signal Processing, 5 (1994), 231-243. doi: 10.1007/BF00980707.
[16]	P. Weiner, "Muldimensional Convolutional Codes,'' Ph.D thesis, University of Notre Dame, USA, 1998.
[17]	X.-L. Zhou and Y. Hu, Multilength two-dimensional codes for optical cdma system, Optoelectronics Letters, 1 (2005), 232-234. doi: 10.1007/BF03033851.

show all references

References:

[1]	E. Fornasini and G. Marchesini, Structure and properties of two-dimensional systems, in "Multidimensional Systems, Techniques and Applications'' (ed. S.G. Tzafestas), (1986), 37-88.
[2]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes, IEEE Trans. Inf. Th., 40 (1994), 1068-1082. doi: 10.1109/18.335967.
[3]	H. Gluesing-Luersen, J. Rosenthal and P. A. Weiner, Duality between mutidimensinal convolutional codes and systems, in "Advances in Mathematical Systems Theory, a Volume in Honor of Diedrich Hinrichsen'' (eds. F. Colonius, U. Helmke, F. Wirth and D. Prtzel-Wolters), Birkhuser, Boston, (2000), 135-150.
[4]	J. Justesen and S. Forchhammer, Two dimensional information theory and coding. With applications to graphics data and high-density storage media, Cambridge University Press, Cambridge, (2010), 171.
[5]	T. Kailath, "Linear Systems,'' Prentice Hall, Englewood Cliffs, NJ, 1980.
[6]	B. Kitchens, Multidimensional convolutional codes, SIAM J. Discrete Math., 15 (2002), 367-381. doi: 10.1137/S0895480100378495.
[7]	B. C. Lévy, "2-D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,'' Ph.D thesis, Stanford University, USA, 1981.
[8]	R. G. Lobo, "On Locally Invertible Encoders and Muldimensional Convolutional Codes,'' Ph.D thesis, North Carolina State University, USA, 2006.
[9]	P. Rocha, "Structure and Representation of 2-D Systems,'' Ph.D thesis, Groningen University, Holland, 1990.
[10]	J. Rosenthal, J. M. Schumacher and E. V. York, On behaviors and convolutional codes, IEEE Trans. Inf. Th., 42 (1996), 1881-1891. doi: 10.1109/18.556682.
[11]	J. Rosenthal and R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput., 10 (1999), 15-32. doi: 10.1007/s002000050120.
[12]	J. Rosenthal and E. V. York, BCH convolutional codes, IEEE Trans. Inf. Th., 45 (1999), 1833-1844. doi: 10.1109/18.782104.
[13]	J. Singh and M. L. Singh, A new family of two-dimensional codes for optical CDMA systems, Optik - International Journal for Light and Electron Optics, 120 (2009), 959-962.
[14]	J. Swartz, T. Pavlidis and Y. P. Wang, Information encoding with two-dimensional bar codes, IEEE Computer Society, 25 (1992), 18-28.
[15]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes: an algebraic approach, Multidimensional Systems and Signal Processing, 5 (1994), 231-243. doi: 10.1007/BF00980707.
[16]	P. Weiner, "Muldimensional Convolutional Codes,'' Ph.D thesis, University of Notre Dame, USA, 1998.
[17]	X.-L. Zhou and Y. Hu, Multilength two-dimensional codes for optical cdma system, Optoelectronics Letters, 1 (2005), 232-234. doi: 10.1007/BF03033851.

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