Raquel Pinto

In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered.

Maximum-distance separable (MDS) convolutional codes are characterized by the property that their free distance reaches the generalized Singleton bound. In this paper, new criteria to construct MDS convolutional codes are presented. These codes also possess optimal first (reverse) column distances. The new criteria allow to relate the construction of MDS convolutional codes to those of reverse superregular Toeplitz matrices. Moreover, using the new criteria as well as the help of computer search, examples for MDS convolutional codes over small finite fields are given.

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in [ ], [ ]. Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control 55(11) (2009) 4890–4897] that, contrary to what happens for codes over [ ], where [ ] is a field, when dealing with [ ] there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called [ ]-encoder. In this paper we solve this conjecture for a class of [ ] convolutional codes over [ ] and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic [ ]-encoder.

Two-dimensional (2D) convolutional codes are a generalization of (1D) convolutional codes, which are suitable for transmission over an erasure channel. In this paper, we present a decoding algorithm for 2D convolutional codes over such a channel by reducing the decoding process to several decoding steps applied to 1D convolutional codes. Moreover, we provide constructions of 2D convolutional codes that are specially tailored to our decoding algorithm.

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable.

Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain rate and a certain degree. We then present two novel constructions that fulfill these conditions by considering two types of superregular matrices.