Abstract
Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.
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This work was supported in part by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.
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Communicated by A. Winterhof.
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Napp, D., Pinto, R. & Toste, M. On MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) . Des. Codes Cryptogr. 83, 101–114 (2017). https://doi.org/10.1007/s10623-016-0204-9
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DOI: https://doi.org/10.1007/s10623-016-0204-9