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In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤ p r , ℤ p r ((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over ((d)) , where is a field, when dealing with ℤ p r ((d)) 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 p -encoder. In this paper we solve this conjecture for a class of (2 , 1) convolutional codes over ℤ p 2 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 p -encoder. [ABSTRACT FROM AUTHOR]

Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate 1 / 2 and degree 5 over the field 1 1 . [ABSTRACT FROM AUTHOR]

Maximum distance profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of $p$ -encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

A convolutional code C over Z p r [ D ] is a Z p r [ D ] -submodule of Z p r n [ D ] where Z p r [ D ] stands for the ring of polynomials with coefficients in Z p r . In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w ∈ C when some of its coefficients have been erased. We do that using the p -adic expansion of w and particular representations of the parity-check polynomial matrix of the code. From these matrix polynomial representations we recursively select certain equations that w must satisfy and have only coefficients in the field p r − 1 Z p r . We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w , that is, a list with the closest codewords to w. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate k/n and degree \delta , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate k/n and degree \delta if k {\nmid } \delta , or n \geq k (({(({\delta }/{k}) + 1)(({\delta }/{k}) + 2)})/{2})$ if $k \mid \delta $ , by presenting a concrete constructive procedure. [ABSTRACT FROM AUTHOR]