The dual of convolutional codes over Zpr\mathbb{Z}_{p^r}

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.Comment: submitte

Propriedades das distâncias dos códigos convolucionais sobre Z pr

Doutoramento em Matemática e AplicaçõesNesta 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, weIn 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

O poder das notícias de crime: imprensa versus televisão

A presente investigação centra-se em saber qual dos dois meios de comunicação escolhidos, nomeadamente, o jornal e a televisão tem mais impacto nas pessoas quando transmite uma notícia de crime, e perceber quais são os fatores que contribuem para isso. Como fio condutor desta investigação foram tratados vários temas importantes até chegar ao estudo prático. A evolução dos media, a criminalidade em Portugal nos últimos onze anos, a relação entre violência e a criminalidade com a sociedade contemporânea, a imagem nos meios de comunicação, bem como a relação entre os media e a sociedade e, os seus efeitos na mesma, foram os assuntos tratados na parte teórica deste trabalho académico. Entre as várias conclusões obtidas nesta dissertação de mestrado, verificou-se que a evolução dos meios de comunicação foi um marco no desenvolvimento da vida do ser humano; que a criminalidade em Portugal diminuiu em 2011 face a 2010, que a sociedade atual “sofre” com os vários efeitos dos media, como é o caso da manipulação e violência das imagens; que a televisão é o meio de comunicação que causa mais impacto nas pessoas quando transmite notícias de crime, e que o jornal é o mais realista e o mais verdadeiro quando transmite as essas notícias.The present investigation is focused in discovering which media, namely, the news paper or television, causes more impact on the public when transmitting the news of a crime and what are the contributor factor for that. As a thrust for this investigation it was studied several important points until the practical study. The media evolution, the criminality rate in Portugal for the past 11 years, the relation between violence and contemporary's society's criminality, media's image, as well as the relation between the media and society, and its effects on the second, were all subjects of study on the theoretical part of this academic project. Among the various conclusions obtained in this masters dissertation, it is showed that the evolution of the media was key for the human being's development; that criminality rate in Portugal lowered from 2010 to 2011, that today's society "suffers" with the various effects from the media, like the manipulation and violence of images; that television is the type of media that impacts the most on the public when transmitting crime related news, but that news papers are more realistic and true when transmitting the same news

Column distances of convolutional codes over Z_p^r

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 Zpr is presented.publishe

Life history of Colpomenia sinuosa (Scytosiphonaceae, Phaeophyceae) in the Azores.

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

On duals and parity-checks of convolutional codes over Z p r

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^{\perp} . We use these results to provide a constructive algorithm to build an explicit generator matrix of C^{\perp}. Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C.publishe

On MDS convolutional codes over Z_p^r

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 Zpr 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 Zpr 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 Zpr for any given set of parameters

A matrix based list decoding algorithm for linear codes over integer residue rings

In this paper we address the problem of list decoding of linear codes over an integer residue ring Zq, where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Zpr , called the standard form, and the p-adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Zpr . From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p^{r−1}Zpr . This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal.publishe