Decoding of Convolutional Codes over the Erasure Channel

In this paper we study the decoding capabilities of convolutional codes over the erasure channel. Of special interest will be maximum distance profile (MDP) convolutional codes. These are codes which have a maximum possible column distance increase. We show how this strong minimum distance condition of MDP convolutional codes help us to solve error situations that maximum distance separable (MDS) block codes fail to solve. Towards this goal, we define two subclasses of MDP codes: reverse-MDP convolutional codes and complete-MDP convolutional codes. Reverse-MDP codes have the capability to recover a maximum number of erasures using an algorithm which runs backward in time. Complete-MDP convolutional codes are both MDP and reverse-MDP codes. They are capable to recover the state of the decoder under the mildest condition. We show that complete-MDP convolutional codes perform in certain sense better than MDS block codes of the same rate over the erasure channel.Comment: 18 pages, 3 figures, to appear on IEEE Transactions on Information Theor

Maximum distance separable 2D convolutional codes

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 $delta$ that reach such bound when $n geq k (({(lfloor ({delta }/{k}) rfloor + 2)(lfloor ({delta }/{k}) rfloor + 3)})/{2})$ 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

Convolutional Goppa Codes

We define Convolutional Goppa Codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some Maximum-Distance Separable (MDS) convolutional codes.Comment: 8 pages, submitted to IEEE Trans. Inform. Theor

Systematic maximum sum rank codes

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.publishe