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1. Nascimento, Ruth. Códigos de peso constante.

Degree: PhD, Matemática, 2014, University of São Paulo

URL: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-07032015-113005/ ;

Sejam F_q um corpo finito com q elementos, e C_n um grupo cíclico de n elementos com mdc(q,n) = 1. Iniciamos nosso trabalho inspirados nos resultados de Vega, estabelecendo condições para que um código de F_qC_n tenha peso constante. Com tal resultado concluímos que um código de peso constante em F_qC_n é da forma {rg^ie | r em F_q, i variando de 0 a n}. A partir disto, determinamos a quantidade de códigos de peso constante de F_qC_n, e construímos exemplos de códigos de dois pesos em F_q(C_n X C_n). Em seguida, estabelecemos sob quais condições um código em F_qA, para A um grupo abeliano finito, tem peso constante. Analisamos também os códigos de peso constante em RG, quando R um anel de cadeia finito e C_n é um grupo cíclico de n elementos com mdc(n,q) = 1. Além disso, analisamos o caso em que os elementos de um ideal de RA, para R um domínio de integridade infinito e A um grupo abeliano finito têm peso constante.

Let F_q be a field with q elements, C_n be a cyclic group of order n and suppose that gcd(q,n) = 1. In this work conditions are given to ensure that a code in F_qC_n is a one weight code, inspired in the work of Vega. As a consequence of this result we showed that a one weight code in F_qC_n is of the form {rg^ie | r in F_q, i between 0 and n}. With this, we determined the number of one weight codes in F_qC_n, and constructed examples of two weight codes in F_q(C_n X C_n). After this, we gave conditions to ensure that a code had constant weight in F_qA, for A a finite abelian group. We also analyzed the one weight codes in RG, R a chain ring and C_n a cyclic group with n elements with gcd(n,q) = 1. Moreover, we analyzed the case when the elements of an ideal in RA, for R an infinite integral domain and A a finite abelian group, have constant weight.

Subjects/Keywords: Abelian group; Anéis de cadeia; Anéis de grupo; Chain ring; Códigos de peso constante; Cyclic group; Group rings; Grupo abeliano; Grupo cíclico; One weight codes

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Nascimento, R. (2014). Códigos de peso constante. (Doctoral Dissertation). University of São Paulo. Retrieved from http://www.teses.usp.br/teses/disponiveis/45/45131/tde-07032015-113005/ ;

Chicago Manual of Style (16^{th} Edition):

Nascimento, Ruth. “Códigos de peso constante.” 2014. Doctoral Dissertation, University of São Paulo. Accessed December 03, 2020. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-07032015-113005/ ;.

MLA Handbook (7^{th} Edition):

Nascimento, Ruth. “Códigos de peso constante.” 2014. Web. 03 Dec 2020.

Vancouver:

Nascimento R. Códigos de peso constante. [Internet] [Doctoral dissertation]. University of São Paulo; 2014. [cited 2020 Dec 03]. Available from: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-07032015-113005/ ;.

Council of Science Editors:

Nascimento R. Códigos de peso constante. [Doctoral Dissertation]. University of São Paulo; 2014. Available from: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-07032015-113005/ ;

2. Krithivasan, Dinesh. Algebraic Structures for Multi-Terminal Communication Systems.

Degree: PhD, Electrical Engineering: Systems, 2010, University of Michigan

URL: http://hdl.handle.net/2027.42/75917

We study a distributed source coding problem with multiple encoders, a central decoder and a joint distortion criterion. The encoders do not communicate with each other. The encoders observe correlated sources which they quantize and communicate noiselessly to a central decoder which is interested in minimizing a joint distortion criterion that depends on the sources and the reconstruction. We are interested in characterizing an inner bound to the optimal rate-distortion region. We first consider a special case where the sources are jointly Gaussian and the decoder wants to reconstruct a linear function of the sources under mean square error distortion. We demonstrate a coding scheme involving nested lattice codes that reconstructs the linear function by encoding in such a fashion that the decoder is able to reconstruct the function directly. For certain source distributions, this approach yields a larger rate-distortion region compared to when the decoder reconstructs lossy versions of the sources first and then estimates the function from them. We then
extend this approach to the case of reconstructing a linear function of an arbitrary
number of jointly Gaussian sources. Next, we consider the general distributed source coding problem with discrete sources. This formulation includes as a special case many famous distributed source coding problems. We present a new achievable rate-distortion region for this problem based on “good” structured nested random codes built over abelian groups. We demonstrate rate gains for this problem over traditional coding schemes using unstructured random codes. For certain sources and distortion functions, the new rate region is strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for the problem till now. Further, there is no known way of achieving these rate gains without exploiting the structure of the coding scheme. Achievable performance limits for single-user source coding using abelian group codes are also obtained as corollaries of the main coding theorem. Our results also imply that nested linear codes achieve the Shannon rate-distortion bound in the single-user setting. Finally, we conclude by outlining some future research directions.
*Advisors/Committee Members: Sadanandarao, Sandeep P. (committee member), Anastasopoulos, Achilleas (committee member), Griess Jr., Robert L. (committee member), Neuhoff, David L. (committee member).*

Subjects/Keywords: Information Theory; Distributed Source Coding; Lattice Coding; Abelian Group Codes; Structured Codes; Electrical Engineering; Engineering

…*abelian*
*group* *codes* are also obtained as corollaries of the main coding theorem. Our results… …*abelian* *group*
*codes*. As a further corollary, we show that nested linear *codes* (built over… …3.8.1 Lossless Source Coding using *Group* *Codes* . . . . . . . . . .
3.8.2 Lossy Source Coding… …using *Group* *Codes* . . . . . . . . . . . .
3.8.3 Nested Linear *Codes*… …126
B.1
B.2
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Good *Group* Channel *Codes* .
Good *Group* Source *Codes* . .
Good…

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Krithivasan, D. (2010). Algebraic Structures for Multi-Terminal Communication Systems. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/75917

Chicago Manual of Style (16^{th} Edition):

Krithivasan, Dinesh. “Algebraic Structures for Multi-Terminal Communication Systems.” 2010. Doctoral Dissertation, University of Michigan. Accessed December 03, 2020. http://hdl.handle.net/2027.42/75917.

MLA Handbook (7^{th} Edition):

Krithivasan, Dinesh. “Algebraic Structures for Multi-Terminal Communication Systems.” 2010. Web. 03 Dec 2020.

Vancouver:

Krithivasan D. Algebraic Structures for Multi-Terminal Communication Systems. [Internet] [Doctoral dissertation]. University of Michigan; 2010. [cited 2020 Dec 03]. Available from: http://hdl.handle.net/2027.42/75917.

Council of Science Editors:

Krithivasan D. Algebraic Structures for Multi-Terminal Communication Systems. [Doctoral Dissertation]. University of Michigan; 2010. Available from: http://hdl.handle.net/2027.42/75917