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, (1986), 37. Google Scholar
[2]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes,, IEEE Trans. Inf. Th., 40 (1994), 1068. doi: 10.1109/18.335967. Google Scholar
[3]	H. Gluesing-Luersen, J. Rosenthal and P. A. Weiner, Duality between mutidimensinal convolutional codes and systems,, in, (2000), 135. Google Scholar
[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, (2010). Google Scholar
[5]	T. Kailath, "Linear Systems,'', Prentice Hall, (1980). Google Scholar
[6]	B. Kitchens, Multidimensional convolutional codes,, SIAM J. Discrete Math., 15 (2002), 367. doi: 10.1137/S0895480100378495. Google Scholar
[7]	B. C. Lévy, "2-D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,'', Ph.D thesis, (1981). Google Scholar
[8]	R. G. Lobo, "On Locally Invertible Encoders and Muldimensional Convolutional Codes,'', Ph.D thesis, (2006). Google Scholar
[9]	P. Rocha, "Structure and Representation of 2-D Systems,'', Ph.D thesis, (1990). Google Scholar
[10]	J. Rosenthal, J. M. Schumacher and E. V. York, On behaviors and convolutional codes,, IEEE Trans. Inf. Th., 42 (1996), 1881. doi: 10.1109/18.556682. Google Scholar
[11]	J. Rosenthal and R. Smarandache, Maximum distance separable convolutional codes,, Appl. Algebra Engrg. Comm. Comput., 10 (1999), 15. doi: 10.1007/s002000050120. Google Scholar
[12]	J. Rosenthal and E. V. York, BCH convolutional codes,, IEEE Trans. Inf. Th., 45 (1999), 1833. doi: 10.1109/18.782104. Google Scholar
[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. Google Scholar
[14]	J. Swartz, T. Pavlidis and Y. P. Wang, Information encoding with two-dimensional bar codes,, IEEE Computer Society, 25 (1992), 18. Google Scholar
[15]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes: an algebraic approach,, Multidimensional Systems and Signal Processing, 5 (1994), 231. doi: 10.1007/BF00980707. Google Scholar
[16]	P. Weiner, "Muldimensional Convolutional Codes,'', Ph.D thesis, (1998). Google Scholar
[17]	X.-L. Zhou and Y. Hu, Multilength two-dimensional codes for optical cdma system,, Optoelectronics Letters, 1 (2005), 232. doi: 10.1007/BF03033851. Google Scholar

show all references

References:

[1]	E. Fornasini and G. Marchesini, Structure and properties of two-dimensional systems,, in, (1986), 37. Google Scholar
[2]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes,, IEEE Trans. Inf. Th., 40 (1994), 1068. doi: 10.1109/18.335967. Google Scholar
[3]	H. Gluesing-Luersen, J. Rosenthal and P. A. Weiner, Duality between mutidimensinal convolutional codes and systems,, in, (2000), 135. Google Scholar
[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, (2010). Google Scholar
[5]	T. Kailath, "Linear Systems,'', Prentice Hall, (1980). Google Scholar
[6]	B. Kitchens, Multidimensional convolutional codes,, SIAM J. Discrete Math., 15 (2002), 367. doi: 10.1137/S0895480100378495. Google Scholar
[7]	B. C. Lévy, "2-D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,'', Ph.D thesis, (1981). Google Scholar
[8]	R. G. Lobo, "On Locally Invertible Encoders and Muldimensional Convolutional Codes,'', Ph.D thesis, (2006). Google Scholar
[9]	P. Rocha, "Structure and Representation of 2-D Systems,'', Ph.D thesis, (1990). Google Scholar
[10]	J. Rosenthal, J. M. Schumacher and E. V. York, On behaviors and convolutional codes,, IEEE Trans. Inf. Th., 42 (1996), 1881. doi: 10.1109/18.556682. Google Scholar
[11]	J. Rosenthal and R. Smarandache, Maximum distance separable convolutional codes,, Appl. Algebra Engrg. Comm. Comput., 10 (1999), 15. doi: 10.1007/s002000050120. Google Scholar
[12]	J. Rosenthal and E. V. York, BCH convolutional codes,, IEEE Trans. Inf. Th., 45 (1999), 1833. doi: 10.1109/18.782104. Google Scholar
[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. Google Scholar
[14]	J. Swartz, T. Pavlidis and Y. P. Wang, Information encoding with two-dimensional bar codes,, IEEE Computer Society, 25 (1992), 18. Google Scholar
[15]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes: an algebraic approach,, Multidimensional Systems and Signal Processing, 5 (1994), 231. doi: 10.1007/BF00980707. Google Scholar
[16]	P. Weiner, "Muldimensional Convolutional Codes,'', Ph.D thesis, (1998). Google Scholar
[17]	X.-L. Zhou and Y. Hu, Multilength two-dimensional codes for optical cdma system,, Optoelectronics Letters, 1 (2005), 232. doi: 10.1007/BF03033851. Google Scholar

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