Composition codes

Ettore  Fornasini; Telma  Pinho; Raquel  Pinto; Paula  Rocha

doi:10.3934/amc.2016.10.163

Article Contents

2016, Volume 10, Issue 1: 163-177. Doi: 10.3934/amc.2016.10.163

This issue Previous Article Further results on fibre products of Kummer covers and curves with many points over finite fields Next Article Decoding of $2$D convolutional codes over an erasure channel

Composition codes

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

Received: November 30, 2014

Revised: June 30, 2015

Published: January 2016

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 94B10, 93C35; Secondary: 93B25, 93B20.

Citation:

Related Papers

Cited by

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), Springer, 1975, 64-82.
[3]	E. Fornasini and R. Pinto, Matrix fraction descriptions in convolutional coding, Linear Algebra Appl., 392 (2004), 119-158.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), 1068-1082.doi: 10.1109/18.335967.
[5]	G. Forney, Convolutional Codes I: Algebraic structure, IEEE Trans. Inf. Theory, 16 (1970), 720-738. Correction, Ibid., 17 (1971), 360.
[6]	G. Forney, Structural analysis of convolutional codes via dual codes, IEEE Trans. Inf. Theory, 19 (1973), 512-518.
[7]	B. Levy, 2D Polynomial and Rational Matrices, and their Applications for the Modeling of 2-D Dynamical Systems, Ph.D thesis, Stanford University, USA, 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), 292-296.doi: 10.1109/TCS.1987.1086219.
[9]	T. Pinho, Minimal State-Space Realizations of 2D Convolutional Codes, Ph.D thesis, Univ. Aveiro, Portugal, 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), 241-250.doi: 10.1007/s10623-012-9768-1.
[11]	P. Rocha, Representation of noncausal 2D systems, in New Trends in Systems Theory, Birkhäuser, 1991, 630-635.
[12]	R. P. Roesser, A Discrete State-Space Model for Linear Image Processing, IEEE Trans. Automat. Control, 20 (1975), 1-10.
[13]	M. E. Valcher and E. Fornasini, On 2D finite support convolutional codes, Multidim. Syst. Signal Proc., 5 (1994), 231-243.doi: 10.1007/BF00980707.
[14]	P. A. Weiner, Multidimensional Convolutional Codes, Ph.D thesis, Univ. Notre Dame, USA, 1998.
[15]	J. C. Willems, Models for dynamics, in Dynamics Reported (eds. U. Kirchgraber and H.O. Walther), John Wiley Sons Ltd., 1989, 171-269.