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Decoding of $2$D convolutional codes over an erasure channel

February 2016, 10(1): 163-177. doi: 10.3934/amc.2016.10.163

Composition codes

Ettore Fornasini ^1, , Telma Pinho ^2, , Raquel Pinto ^2, and Paula Rocha ^3,

1.	Department of Information Engineering, University of Padua, Italy
2.	CIDMA, Department of Mathematics, University of Aveiro, Portugal, Portugal
3.	SYSTEC, Faculty of Engineering, University of Porto, Portugal

Received December 2014 Revised July 2015 Published March 2016

Abstract
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In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders $G(d_1,d_2)$ that can be decomposed as the product of two 1D encoders, i.e., $ G(d_1,d_2)=G_2(d_2)G_1(d_1)$. Taking into account this decomposition, we obtain syndrome formers of the code directly from $G_1(d_1)$ and $ G_2(d_2)$, in case $G_1(d_1)$ and $ G_2(d_2)$ are right prime. Moreover we consider 2D state-space realizations by means of a separable Roesser model of the encoders and syndrome formers of a composition code and we investigate the minimality of such realizations. In particular, we obtain minimal realizations for composition codes which admit an encoder $G(d_1,d_2)=G_2(d_2)G_1(d_1)$ with $G_2(d_2)$ a systematic 1D encoder. Finally, we investigate the minimality of 2D separable Roesser state-space realizations for syndrome formers of these codes.

Keywords: Encoders and syndrome formers, 2D composition codes, minimal 2D state-space models..

Mathematics Subject Classification: Primary: 94B10, 93C35; Secondary: 93B25, 93B2.

Citation: Ettore Fornasini, Telma Pinho, Raquel Pinto, Paula Rocha. Composition codes. Advances in Mathematics of Communications, 2016, 10 (1) : 163-177. doi: 10.3934/amc.2016.10.163

References:

[1]	S. Attasi, Systèmes linéaires homogènes à deux indices,, in Rapport Laboria, (1973).
[2]	E. Fornasini and G. Marchesini, Algebraic realization theory of two-dimensional filters,, in Variable Structure Systems with Application to Economics and Biology* (eds. A. Ruberti and R. Mohler)*, (1975), 64.
[3]	E. Fornasini and R. Pinto, Matrix fraction descriptions in convolutional coding,, Linear Algebra Appl., 392 (2004), 119. doi: 10.1016/j.laa.2004.06.007.
[4]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes,, IEEE Trans. Inf. Theory, 40 (1994), 1068. doi: 10.1109/18.335967.
[5]	G. Forney, Convolutional Codes I: Algebraic structure,, IEEE Trans. Inf. Theory, 16 (1970), 720.
[6]	G. Forney, Structural analysis of convolutional codes via dual codes,, IEEE Trans. Inf. Theory, 19 (1973), 512.
[7]	B. Levy, 2D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,, Ph.D thesis, (1981).
[8]	T. Lin, M. Kawamata and T. Higuchi, Decomposition of 2-D separable-denominator systems: Existence, uniqueness, and applications,, IEEE Trans. Circ. Syst., 34 (1987), 292. doi: 10.1109/TCS.1987.1086219.
[9]	T. Pinho, Minimal State-Space Realizations of 2D Convolutional Codes,, Ph.D thesis, (2014).
[10]	T. Pinho, R. Pinto and P. Rocha, Realization of 2D convolutional codes of rate $\frac1n$ by separable Roesser models,, Des. Codes Crypt., 70 (2014), 241. doi: 10.1007/s10623-012-9768-1.
[11]	P. Rocha, Representation of noncausal 2D systems,, in New Trends in Systems Theory, (1991), 630.
[12]	R. P. Roesser, A Discrete State-Space Model for Linear Image Processing,, IEEE Trans. Automat. Control, 20 (1975), 1.
[13]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes,, Multidim. Syst. Signal Proc., 5 (1994), 231. doi: 10.1007/BF00980707.
[14]	P. A. Weiner, Multidimensional Convolutional Codes,, Ph.D thesis, (1998).
[15]	J. C. Willems, Models for dynamics,, in Dynamics Reported* (eds. U. Kirchgraber and H.O. Walther)*, (1989), 171.

show all references

References:

[1]	S. Attasi, Systèmes linéaires homogènes à deux indices,, in Rapport Laboria, (1973).
[2]	E. Fornasini and G. Marchesini, Algebraic realization theory of two-dimensional filters,, in Variable Structure Systems with Application to Economics and Biology* (eds. A. Ruberti and R. Mohler)*, (1975), 64.
[3]	E. Fornasini and R. Pinto, Matrix fraction descriptions in convolutional coding,, Linear Algebra Appl., 392 (2004), 119. doi: 10.1016/j.laa.2004.06.007.
[4]	E. Fornasini and M. E. Valcher, Algebraic aspects of two-dimensional convolutional codes,, IEEE Trans. Inf. Theory, 40 (1994), 1068. doi: 10.1109/18.335967.
[5]	G. Forney, Convolutional Codes I: Algebraic structure,, IEEE Trans. Inf. Theory, 16 (1970), 720.
[6]	G. Forney, Structural analysis of convolutional codes via dual codes,, IEEE Trans. Inf. Theory, 19 (1973), 512.
[7]	B. Levy, 2D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems,, Ph.D thesis, (1981).
[8]	T. Lin, M. Kawamata and T. Higuchi, Decomposition of 2-D separable-denominator systems: Existence, uniqueness, and applications,, IEEE Trans. Circ. Syst., 34 (1987), 292. doi: 10.1109/TCS.1987.1086219.
[9]	T. Pinho, Minimal State-Space Realizations of 2D Convolutional Codes,, Ph.D thesis, (2014).
[10]	T. Pinho, R. Pinto and P. Rocha, Realization of 2D convolutional codes of rate $\frac1n$ by separable Roesser models,, Des. Codes Crypt., 70 (2014), 241. doi: 10.1007/s10623-012-9768-1.
[11]	P. Rocha, Representation of noncausal 2D systems,, in New Trends in Systems Theory, (1991), 630.
[12]	R. P. Roesser, A Discrete State-Space Model for Linear Image Processing,, IEEE Trans. Automat. Control, 20 (1975), 1.
[13]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes,, Multidim. Syst. Signal Proc., 5 (1994), 231. doi: 10.1007/BF00980707.
[14]	P. A. Weiner, Multidimensional Convolutional Codes,, Ph.D thesis, (1998).
[15]	J. C. Willems, Models for dynamics,, in Dynamics Reported* (eds. U. Kirchgraber and H.O. Walther)*, (1989), 171.

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