Talk
eSeminar: MDP Convolutional Codes over \(\mathbb{Z}_{p^r}\)
Talk by Anina Gruica
Date: 08.04.20 Time: 15.00 - 16.00 Room:
Maximum distance profile (MDP) convolutional codes are characterized by the property that their column distances are maximal among all codes of the same rate and degree, and they have the ability to correct a maximal number of errors per time interval. Over finite fields, MDP convolutional codes have been studied widely and several constructions have been presented. Most of these constructions are based on a well-known algebraic characterization of their truncated sliding generator matrices. This characterization gives a criterion, which says that a left-prime polynomial matrix of fixed size is a generator matrix of an MDP convolutional code, if all the full-size minors of the truncated sliding generator matrix, that are nontrivially zero, are nonzero. In this presentation, my main focus lies on MDP convolutional codes over \(\mathbb{Z}_{p^r}\). There are several problems we encounter when defining convolutional codes over \(\mathbb{Z}_{p^r}\), one of them being that submodules of \(\mathbb{Z}_{p^r}^n[z]\) do not necessarily have a basis. For this reason, I will present the necessary theory on submodules of \(\mathbb{Z}_{p^r}^n[z]\), in order to be able to define an alternative for a basis, which is called \(p\)-basis. With this \(p\)-basis, we get an analogue of a generator matrix and we can analyse column distance properties of convolutional codes over \(\mathbb{Z}_{p^r}\) linked to their truncated sliding generator matrices. Not very long ago upper bounds for column distances of convolutional codes over \(\mathbb{Z}_{p^r}\) have been derived. Using these upper bounds and the concept of \(p\)-encoders we give an analogue of the characterization of MDP convolutional codes over finite fields for MDP convolutional codes over \(\mathbb{Z}_{p^r}\). In particular, we present a construction of MDP convolutional codes over \(\mathbb{Z}_{p^r}\) based on this characterization and the novel notion of \(p\)-superregular matrices. Click here to go to the live streaming of the talk.