Composition codes
Ettore Fornasini Telma Pinho Raquel Pinto Paula Rocha
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.

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