Construction and decoding of convolutional codes over the erasure channel
Final Report Abstract
During the first year of my research fellowship, I managed to estimate the necessary field size and the probability for MDP and complete MDP convolutional codes. Moreover, I introduced j-complete MDP convolutional codes, developed a low delay decoding algorithm for convolutional codes and showed that j-complete MDP convolutional codes are optimal for this low delay decoding. Besides, we obtained the minimal possible field size and corresponding constructions for j-complete MDP convolutional codes with certain parameters and were able to classify all complete MDP convolutional codes over a field of minimal size for these parameters. Furthermore, we provided a construction for MDP convolutional codes using superregular matrices and developed decoding algorithms for 2-dimensional convolutional codes and for convolutional codes over integer residue rings. In the second year of my project, we developed a decoding algorithm for convolutional codes using the linear-systems representation and provided conditions on the corresponding matrices (A,B,C,D) in order to get a good performance with this algorithm. Moreover, we derived a matrix theoretic result on the left primeness of certain polynomial matrices and applied this result to simplify the most commonly used criterion to check whether a convolutional code is MDP. Finally, we were working on the construction of LDPC convolutional codes using certain combinatorial objects called difference triangle sets.
Publications
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Complete j-MDP convolutional codes, IEEE Transactions on Information Theory, 2020
Almeida, Paulo J. & Lieb, Julia
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Convolutional Codes, in A Concise Encyclopedia of Coding Theory (eds. Huffman, C; Kim, J.; Sole, P.), CRC Press, 2020
Lieb, J.; Pinto, R.; Rosenthal, J.
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List decoding of Convolutional Codes over integer residue rings, 2020
Lieb, J.; Napp, D.; Pinto, R.
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On the left primeness of some polynomial matrices with applications to convolutional codes, Journal of Algebra and Its Applications, 2020.
Alfarano, Gianira N. & Lieb, Julia
