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Maximum distance profile codes over finite nonbinary 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 Z_{p^r} is presented. published

Rosenthal et al. introduced and thoroughly studied the notion of Maximum Distance Profile (MDP) convolutional codes over (non-binary) finite fields refining the classical notion of optimum distance profile, see for instance [18, p.164]. These codes have the property that their column distances are maximal among all codes of the same rate and the same degree. In this paper we aim at studying this fundamental notion in the context of convolutional codes over a finite ring. We extensively use the notion of p-encoder to present upper-bounds on the column distances which allow to introduce the notion of MDP in the context of finite rings. A constructive method for (non necessarily free) MDP convolutional codes over Z p r is presented., Submitted. arXiv admin note: text overlap with arXiv:1601.04507

Submitted by Diego Napp Avelli (diego@ua.pt) on 2016-10-03T16:08:49Z No. of bitstreams: 1 art%3A10.1007%2Fs10623-016-0204-9.pdf: 482251 bytes, checksum: 4d279a78795e1ea447037481b6ab44c1 (MD5) Approved for entry into archive by Rita Goncalves(ritaisabel@ua.pt) on 2016-10-10T13:58:00Z (GMT) No. of bitstreams: 1 art%3A10.1007%2Fs10623-016-0204-9.pdf: 482251 bytes, checksum: 4d279a78795e1ea447037481b6ab44c1 (MD5) Made available in DSpace on 2016-10-10T13:58:00Z (GMT). No. of bitstreams: 1 art%3A10.1007%2Fs10623-016-0204-9.pdf: 482251 bytes, checksum: 4d279a78795e1ea447037481b6ab44c1 (MD5) Previous issue date: 2016-04-15

An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional codes from finite fields to finite rings have attracted much attention in recent years due to fact that they are the most appropriate codes for phase modulation. However convolutional codes over finite rings are more involved and not fully understood. Many results and features that are well-known for convolutional codes over finite fields have not been fully investigated in the context of finite rings. In this paper we focus in one of these unexplored areas, namely, we investigate the dual codes of convolutional codes over finite rings. In particular we study the p-dimension of the dual code of a convolutional code over a finite ring. This contribution can be considered a generalization and an extension, to the rings case, of the work done by Forney and McEliece on the dimension of the dual code of a convolutional code over a finite field., submitted

Doutoramento em Matemática e Aplicações Nesta tese consideramos códigos convolucionais sobre o anel polinomial [ ] r p ′ D , onde p é primo e r é um inteiro positivo. Em particular, focamo-nos no conjunto das palavras de código com suporte finito e estudamos as suas propriedades no que respeita às distâncias. Investigamos as duas propriedades mais importantes dos códigos convolucionais, nomeadamente, a distância livre e a distância de coluna. Começamos por analisar e solucionar o problema de, dado um conjunto de parâmetros, determinar a distância livre máxima possível que um código convolucional sobre [ ] r p ′ D pode atingir. Com efeito, obtemos um novo limite superior para esta distância generalizando os limites obtidos no contexto dos códigos convolucionais sobre corpos finitos. Além disso, mostramos que esse limite é ótimo, no sentido em que não pode ser melhorado. Para tal, apresentamos construções de códigos convolucionais (não necessariamente livres) que permitem atingir esse limite, para um certo conjunto de parâmetros. De acordo com a literatura chamamos a esses códigos MDS. Definimos também distâncias de coluna de um código convolucional. Obtemos limites superiores para as distâncias de coluna e chamamos MDP aos códigos cujas distâncias de coluna atingem estes limites superiores. Além disso, mostramos a existência de códigos MDP. Note-se, porém, que os códigos MDP apresentados não são completamente gerais pois os seus parâmetros devem satisfazer determinadas condições. Finalmente, estudamos o código dual de um código convolucional definido em (( )) r p ′ D . Os códigos duais de códigos convolucionais sobre corpos finitos foram exaustivamente investigados, como é refletido na literatura sobre o tema. Estes códigos são relevantes pois fornecem informação sobre a distribuição dos pesos do código e é neste sentido a inclusão deste assunto no âmbito desta tese. Outra razão importante para o estudo de códigos duais é a sua utilidade para o desenvolvimento de algoritmos de descodificação quando consideramos um erasure channel. Nesta tese são analisadas algumas propriedades fundamentais dos duais. Em particular, mostramos que códigos convolucionais definidos em (( )) r p ′ D admitem uma matriz de paridade. Para além disso, apresentamos um método construtivo para determinar um codificador de um código dual. keywords Convolutional codes, finite rings, free distance, column distance, MDS, MDP, dual code abstract In this thesis we consider convolutional codes over the polynomial ring [ ] r p ′ D , where p is a prime and r is a positive integer. In particular, we focus in the set of finite support codewords and study their distances properties. We investigate the two most important distance properties of convolutional codes, namely, the free distance and the column distance. First we address and fully solve the problem of determining the maximum possible free distance a convolutional code over [ ] r p ′ D can achieve, for a given set of parameters. Indeed, we derive a new upper bound on this distance generalizing the Singleton-type bounds derived in the context of convolutional codes over finite fields. Moreover, we show that such a bound is optimal in the sense that it cannot be improved. To do so we provide concrete constructions of convolutional codes (not necessarily free) that achieve this bound for any given set of parameters. In accordance with the literature we called such codes Maximum Distance Separable (MDS). We define the notion of column distance of a convolutional code. We obtain upper-bounds on the column distances and call Maximum Distance Profile (MDP) the codes that attain the maximum possible column distances. Furthermore, we show the existence of MDP codes. We note however that the MDP codes presented here are not completely general as their parameters need to satisfy certain conditions. Finally, we study the dual code of a convolutional code defined in (( )) r p ′ D . Dual codes of convolutional codes over finite fields have been thoroughly investigated as it is reflected in the large body of literature on this topic. They are relevant as they provide value information on the weight distribution of the code and therefore fit in the scope of this thesis. Another important reason for the study of dual codes is that they can be very useful for the development of decoding algorithms of convolutional codes over the erasure channel. In this thesis some fundamental properties have been analyzed. In particular, we show that convolutional codes defined in (( )) r p ′ D admit a parity-check matrix. Moreover, we In this thesis we consider convolutional codes over the polynomial ring [ ] r p ′ D , where p is a prime and r is a positive integer. In particular, we focus in the set of finite support codewords and study their distances properties. We investigate the two most important distance properties of convolutional codes, namely, the free distance and the column distance. First we address and fully solve the problem of determining the maximum possible free distance a convolutional code over [ ] r p ′ D can achieve, for a given set of parameters. Indeed, we derive a new upper bound on this distance generalizing the Singleton-type bounds derived in the context of convolutional codes over finite fields. Moreover, we show that such a bound is optimal in the sense that it cannot be improved. To do so we provide concrete constructions of convolutional codes (not necessarily free) that achieve this bound for any given set of parameters. In accordance with the literature we called such codes Maximum Distance Separable (MDS). We define the notion of column distance of a convolutional code. We obtain upper-bounds on the column distances and call Maximum Distance Profile (MDP) the codes that attain the maximum possible column distances. Furthermore, we show the existence of MDP codes. We note however that the MDP codes presented here are not completely general as their parameters need to satisfy certain conditions. Finally, we study the dual code of a convolutional code defined in (( )) r p ′ D . Dual codes of convolutional codes over finite fields have been thoroughly investigated as it is reflected in the large body of literature on this topic. They are relevant as they provide value information on the weight distribution of the code and therefore fit in the scope of this thesis. Another important reason for the study of dual codes is that they can be very useful for the development of decoding algorithms of convolutional codes over the erasure channel. In this thesis some fundamental properties have been analyzed. In particular, we show that convolutional codes defined in (( )) r p ′ D admit a parity-check matrix. Moreover, we provide a constructive method to explicitly compute an encoder of the dual code.

Doutoramento em Matemática e Aplicações Submitted by Cristina Santos (cmaria@ua.pt) on 2017-06-26T08:52:33Z No. of bitstreams: 1 TESE Marisa toste.pdf: 666912 bytes, checksum: bcaa48d8e6b5a9fd2a8c891d1e25fd1f (MD5) Made available in DSpace on 2017-06-26T08:52:35Z (GMT). No. of bitstreams: 1 TESE Marisa toste.pdf: 666912 bytes, checksum: bcaa48d8e6b5a9fd2a8c891d1e25fd1f (MD5) Previous issue date: 2016

Copyright © 2003 Phycological Society of America. 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.

Copyright © 2003 Adac. Tous droits réservés. Hidroclathrus clathradus is a common specie on the rocky intertidal shores of the island of São Miguel, Azores, mainly in spring and summer. Fertile saccate thaili appear in spring and disappear from late summer onward. Two types of life history were observed in culture: a direct-type and a heteromorphic, monophasic life history. In the direct type, reproductive cells from erect plants collected in the field developed directly into new erect thaili with plurilocular sporangia. In the heteromorphic life-hystory pattern, plurispores from the plurilocular sporangia of the erect thaili developed into filamentous prostate microthalli. These developed unilocular and plurilocular sporangia in both short-day and long-day conditions at 15-22 ºC. Unispores released from the unlocular sporangia developed into saccate thaili, whereas plurispores gave rise to filamentous prostate microthaili. RÉSUMÉ: Hidroclathrus clathradus est une espèce commune de l'étage littoral sur les côtes rocheuses de l'íle de São Miguel. Açores, particulièrement au printemps et en été. Des boules irrégulières fertiles sont visibles dès le début du printemps jusqu'a la fin de I'été. Deux types de cycle vital ont été observés en culture: un type de développement direct et un autre monophasique et hétéromorphe. Dans le cycle monomorphique, des cellules reproductives des échantillons collectés se développent directement en nouvelles boules avec des sporocystes pluriloculaires. Dans le cycle vital hétéromorphique, des plurispores provenant des sporocystes pluriloculaires des thailes creux se développent en microthalles filamenteux. Ceux-ci donnent naissance à des sporocystes uni- et pluriloculaires à 15-22 ºC, que le jour soir long ou court. Les monospores libérées des sporocystes pluriloculaires donnent des gamétophytes globuleux tandis que les plurispores se développent en microthalles filamenteux.

36th European Marine Biology Symposium. Maó-Minorca, Espanha, 17-22 de Setembro de 2001.

7th International Phycological Congress. Thessaloniki, Grécia, 18-25 de Agosto de 2001.