Construction and decoding of convolutional codes over the erasure channel

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Coding theory is dealing with error correction during data transmission using error correcting codes. The principle of an error correcting code is to add redundancy to a message, in order to make it possible to correct transmission errors or to reconstruct lost data.The aim of the project is to develop new constructions and efficient decoding algorithms for maximum distance profile (MDP) convolutional codes over the erasure channel. When using this channel, a transmitted symbol is either received correctly or gets lost. Moreover, special subclasses of MDP convolutional codes, the reverse-MDP and complete-MDP convolutional codes should be considered. These codes have additional properties, which are of advantage for decoding. The motivation for this project is provided by the necessity for decoding with least possible delay in applications like internet traffic and in particular, video streaming.In the first part of the project, I want to develop general constructions for reverse-MDP and complete-MDP convolutional codes, where for complete-MDP convolutional codes, one has to prove their existence for all code parameters first. Up to now there are only constructions for MDP convolutional codes over fields of large size, which makes decoding algorithms very complex. Therefore, another aim is to develop constructions of MDP convolutional codes for small field sizes and to obtain a bound for the necessary field size such that a MDP convolutional code exists.In the second part of the project, I plan to approach the development of efficient decoding algorithms for MDP convolutional codes over the erasure channel in two ways: firstly, by considering the parity-check matrix and secondly, by using the systems-theoretic representation of a convolutional code. For the second way, I will use that each convolutional code could be described by a linear system over a finite field.Finally, I will investigate if the class of MDP convolutional codes is capable to achieve Shannon capacity over the q-ary erasure channel without memory, i.e. the aim is to find codes with possibly large transmission rates and possibly small error probabilities. When using the above channel, symbols from a finite field with q elements are transmitted and the erasure probability of a symbols is independent of previous symbols.