Abstract
In this paper, we investigate extended row distances of Unit Memory (UM) convolutional codes . In particular, we derive upper and lower bounds for these distances and moreover present a concrete construction of a UM convolutional code that almost achieves the derived upper bounds. The generator matrix of these codes is built by means of a particular class of matrices, called superregular matrices . We actually conjecture that the construction presented is optimal with respect to the extended row distances as it achieves the maximum extended row distances possible. This in particular implies that the upper bound derived is not completely tight. The results presented in this paper further develop the line of research devoted to the distance properties of convolutional codes which has been mainly focused on the notions of free distance and column distance. Some open problems are left for further research.
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Notes
- 1.
Therefore, the degree \(\delta \) of a convolutional code \({\mathscr {C}}\) is the sum of the row degrees of one, and hence any, minimal basic encoder.
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Acknowledgements
The authors would like to thank the reviewer for his/her comments that led to improve the quality of the final version. This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project PEst-UID/MAT/04106/2013.
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Almeida, P., Napp, D., Pinto, R. (2017). On Optimal Extended Row Distance Profile. In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_4
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