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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 ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Colpomenia sinuosa (Mertens ex Roth) Derbès and Solier (Scytosiphonaceae, Phaeophyceae) is a common species on the rocky intertidal shores of the Azores, where reproductive gametophytes occur throughout the year. Life-history studies of this species were carried out in culture, and both sexual and asexual reproduction were observed. Anisogamous gametes fused to form zygotes. The zygotes gave rise to a filamentous prostrate sporophyte generation bearing unilocular sporangia, under both short-day and long-day conditions at 15 and 22°C, and to both unilocular and plurilocular sporangia, under the lower temperature condition. Unispores developed into gametophytes, and plurispores gave rise to filamentous sporophytes. Asexual reproduction was carried out by unfused female gametes and asexual plurispores produced from the same gametophyte. Unfused gametes developed into filamentous prostrate sporophytes producing unilocular sporangia in both culture conditions, and unispores released from the sporangia gave rise to gametophytes. Asexual plurispores from field gametophytes, under both culture conditions, developed directly into new gametophytes. The species exhibited three types of life history: a heteromorphic, diplohaplontic; a heteromorphic, monophasic (both with alternation between the erect and filamentous prostrate thalli); and a monomorphic, monophasic. [ABSTRACT FROM AUTHOR]

Abstract A convolutional code C over Z p r ((D)) is a Z p r ((D)) -submodule of Z p r n ((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. [ABSTRACT FROM AUTHOR]