Resumo 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 Z p r was recently discovered in [26] via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over Z p r from a new perspective. We introduce the notions of p-standard form and roptimal 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 Z p r for any given set of parameters.

A convolutional code C over Z p r ((D)) is a Z p r ((D))-submodule of Z n p r ((D)) that admits a polynomial set of generators, where Z p r ((D)) stands for the ring of (semi-infinity) Laurent series. In this paper we study several structural properties of its dual C ⊥ . We use these results to provide a constructive algorithm to build an explicit generator matrix of C ⊥ . Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C.

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Maximum Distance Profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) Maximum Distance Profile convolutional codes over Zpr is presented. I. INTRODUCTION Massey and Mittelholzer [19] showed that the most appropriate codes for phase modulation are the linear codes over the residue class ring Z M and this class includes the convolutional codes over Z M , where M is a positive integer. Fundamental results of the structural properties of convolutional codes over finite rings can be found, for instance, in [7] and [12]. Fagnani and Zampieri [7] studied the theory of convolutional codes over the ring Z p r in the case when the input sequence space is a free module. The problem of deriving minimal encoders (left prime and row-reduced) was posed by Solé et al. in [26] and solved by Kuijper et al. in [16] and [17] using the concept of minimal p-encoder, which is an extension of the concept of p-basis introduced in [29] to the polynomial context. The search for and design of good convolutional codes over Z p r have been investigated in several works in literature. Unit-memory convolutional codes over Z 4 that give rise to binary trellis codes with high free distances together with several concrete constructions of these codes were reported in [2] and [15]. In [13] two 16-state trellis codes of rate 2/4, again over Z 4 , were found by computer search. Also worth