Input-state-output representations and constructions of finite support 2D convolutional codes
Diego Napp Carmen Perea Raquel Pinto
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
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.
Raquel Pinto Paula Rocha Paolo Vettori
The $4^{th}$International Castle Meeting on Coding Theory and its Applications (4ICMCTA) took place in the Palmela Castle, Portugal, on September 15--18, 2014. It was organized under the auspices of the Research & Development Center for Mathematics and Applications (CIDMA) from the University of Aveiro. Following in the spirit of the previous installments held at La Mota Castle, Spain, in 1999 and 2008, and at Cardona Castle, Spain, in 2011, the meeting has been a good opportunity for communicating new results, exchanging ideas, strengthening international cooperation, and introducing young researchers into the Coding Theory community.

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Decoding of $2$D convolutional codes over an erasure channel
Joan-Josep Climent Diego Napp Raquel Pinto Rita Simões
In this paper we address the problem of decoding $2$D convolutional codes over an erasure channel. To this end we introduce the notion of neighbors around a set of erasures which can be considered an analogue of the notion of sliding window in the context of $1$D convolutional codes. The main idea is to reduce the decoding problem of $2$D convolutional codes to a problem of decoding a set of associated $1$D convolutional codes. We first show how to recover sets of erasures that are distributed on vertical, horizontal and diagonal lines. Finally we outline some ideas to treat any set of erasures distributed randomly on the $2$D plane.
keywords: $2$D finite support convolutional codes $1$D finite support convolutional codes erasure channel.

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