American Institute of Mathematical Sciences

Advanced
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

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

[1]

Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões. Decoding of $2$D convolutional codes over an erasure channel. Advances in Mathematics of Communications, 2016, 10 (1) : 179-193. doi: 10.3934/amc.2016.10.179
[2]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004
[3]

James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445
[4]

Nils Dabrock, Yves van Gennip. A note on "Anisotropic total variation regularized $L^1$-approximation and denoising/deblurring of 2D bar codes". Inverse Problems & Imaging, 2018, 12 (2) : 525-526. doi: 10.3934/ipi.2018022
[5]

Rustum Choksi, Yves van Gennip, Adam Oberman. Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes. Inverse Problems & Imaging, 2011, 5 (3) : 591-617. doi: 10.3934/ipi.2011.5.591
[6]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61
[7]

Mehdi Badra, Takéo Takahashi. Feedback boundary stabilization of 2d fluid-structure interaction systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2315-2373. doi: 10.3934/dcds.2017102
[8]

Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems. Electronic Research Announcements, 2019, 26: 54-71. doi: 10.3934/era.2019.26.005
[9]

Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685
[10]

Ka Kit Tung, Wendell Welch Orlando. On the differences between 2D and QG turbulence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 145-162. doi: 10.3934/dcdsb.2003.3.145
[11]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[12]

Géry de Saxcé, Claude Vallée. Structure of the space of 2D elasticity tensors. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1525-1537. doi: 10.3934/dcdss.2013.6.1525
[13]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
[14]

Bernd Kawohl, Guido Sweers. On a formula for sets of constant width in 2d. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2117-2131. doi: 10.3934/cpaa.2019095
[15]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[16]

José Ignacio Iglesias Curto. Generalized AG convolutional codes. Advances in Mathematics of Communications, 2009, 3 (4) : 317-328. doi: 10.3934/amc.2009.3.317
[17]

Heide Gluesing-Luerssen. On isometries for convolutional codes. Advances in Mathematics of Communications, 2009, 3 (2) : 179-203. doi: 10.3934/amc.2009.3.179
[18]

Quoc T. Luu, Paul DuChateau. The relative biologic effectiveness versus linear energy transfer curve as an output-input relation for linear cellular systems. Mathematical Biosciences & Engineering, 2009, 6 (3) : 591-602. doi: 10.3934/mbe.2009.6.591
[19]

Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011
[20]

Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences