Marcelo Firer - Profile on Academia.edu (original) (raw)

Papers by Marcelo Firer

Espaços poset e o problema da distribuição de pesos

Neste trabalho fazemos uma apresentacao dos espacos poset, introduzidos por Brualdi (1995), apres... more Neste trabalho fazemos uma apresentacao dos espacos poset, introduzidos por Brualdi (1995), apresentamos os conceitos necessarios da teoria de conjuntos parcialmente ordenados e da teoria de codigos. Trabalhamos com uma questao de carater amplo e estrutural deste contexto, o problema da determinacao da ordem atraves da distribuicao de pesos. A distribuicao de pesos e essencialmente o conjunto das cardinalidades das esferas metricas e a pergunta que se coloca e em que medida este invariante determina a metrica em questao. Demonstramos que para as classes de codigos, cadeia, anticadeia, coroa e hierarquico, classes importantes no contexto da teoria de codigos, o problema possui uma resposta positiva e justificamos algumas conjecturas que relacionam este problema ao da reconstrucao de grafos Abstract

This work is about two topics in Coding and Information Theory in hyperbolic spaces: (i) Error pr... more This work is about two topics in Coding and Information Theory in hyperbolic spaces: (i) Error probability of signal transmissions in hyperbolic spaces; (ii) Signal constellations with geometric properties feasible for practical applications in environments with hyperbolic behavior. In the first item, it was developed an upper bound for the error probability of signal transmissions in ndimensional hyperbolic space. It was obtained a class of probability density functions for hyperbolic noise analogous to the Gaussian noise in n-dimensional Euclidean space, that we called "hyperbolic Gaussian noise". In this way, the performance comparison between hyperbolic signal constellations perturbed by this type of noise is reasonable for computation. In addition, signal constellations of hyperbolic M-PSK type are analyzed in terms of performance concerning error probability. Related to the second item it was obtained both geometrically and non-geometrically uniform signal constellations on surfaces constructed through quotients of hyperbolic spaces by discrete groups of isometries. The infinite constellations over non-compact surfaces obtained by this process are similar to chrystallographic group lattices in Euclidean space. The finite non-geometrically uniform constellations over compact surfaces are generated by self-intersection of unique knots over the torus of genus g and are cyclic labeled by the group Z n. In addition, an analysis of transmission in channels perturbed by "hyperbolic Gaussian noise", according to the description of the first item is develloped.

Revista Professor de Matemática On line, 2022

Este texto apresenta uma sequência de atividades didáticas para ser realizada com alunos do Ensin... more Este texto apresenta uma sequência de atividades didáticas para ser realizada com alunos do Ensino Básico, com o objetivo de explorar o funcionamento do brinquedo chamado espirógrafo. Os conteúdos matemáticos utilizados e explorados são elementares (proporcionalidade, mínimo múltiplo comum e máximo divisor comum), e a sequência visa propiciar aos alunos uma descoberta orientada, baseada em observações de fenômenos, e concluída com uma demonstração transformacional.

Poset Codes: Partial Orders, Metrics and Coding Theory

SpringerBriefs in Mathematics, 2018

Basic Concepts of Coding Theory

Poset Codes: Partial Orders, Metrics and Coding Theory, 2018

We briefly introduce some very basic facts about the theory of error correcting codes or coding t... more We briefly introduce some very basic facts about the theory of error correcting codes or coding theory. To be more precise, we introduce those basic facts that are related to or dependent on the concept of metric. Many hypothesis are presented in a simplified form or in a restricted context, except for the concept of metric, which is considered in full generality, not really specified at this point, since this is the main goal of these notes: to show the role that different metrics may have in the context of coding theory and to present some interesting questions and bias to the study of finite metric spaces that arouse from coding theory problems and invariants.

2017 IEEE International Symposium on Information Theory (ISIT), 2017

In this work, we adapt the notion of generalized Hamming weight of block codes to introduce the n... more In this work, we adapt the notion of generalized Hamming weight of block codes to introduce the novel concept of generalized column distances for convolutional codes. This can be considered as an extension of the work done in [18] on the generalized Hamming weights for free distance of convolutional codes. We also introduce the concept of Almost-MDP and Near-MDP convolutional code. The problem of constructing convolutional codes with design generalized column distances remains an interesting open problem that requires further research.

Disjoint Chains with Equal Length: The Niederreiter-Rosenbloom-Tsfasman Metric

Poset Codes: Partial Orders, Metrics and Coding Theory, 2018

The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel... more The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel developments.

Special issue on “Many Faces of Distances”

International Transactions in Operational Research, 2014

ArXiv, 2018

In this work we explore the family of metrics determined by SSS-weights, i.e., non-negative funct... more In this work we explore the family of metrics determined by SSS-weights, i.e., non-negative functions over finite fields that respect the support. First, we introduce a conditional sum of weights and classify those which every set of equivalent weights is closed under such sums. Then, we introduce an structure to represent all decision criteria which allows us to characterize the group of linear isometries for SSS-weights sharing the same equivalence class regarding the decoding criterion.

Caracterização de métricas induzidas por posets hierárquicos

Neste trabalho, utilizando a decomposicao canonica para codigos em espacos munidos de metricas de... more Neste trabalho, utilizando a decomposicao canonica para codigos em espacos munidos de metricas definidas por ordens parciais hierarquicas, sao apresentadas formulas explicitas para os principais parâmetros de um codigo linear (distância minima, raios de empacotamento, de cobertura e de Chebyshev) em termo dos correspondentes na metrica de Hamming. Alem disso, e apresentada uma serie de propriedades que caracterizam as metricas definidas por ordens parciais hierarquicas, sendo algumas das caracteristicas originais e, para todas estas propriedades, apresentamos demonstracoes originais, simples e curtas. Abstract

Poset Metrics

SpringerBriefs in Mathematics, 2018

Generalizations, Variations and Perspectives

In this chapter we present different generalizations and variations of the poset metrics. The app... more In this chapter we present different generalizations and variations of the poset metrics. The approach adopted here is very concise. For each of these generalizations we introduce the proper definitions and only explain the concepts needed to understand the statement of the main results (whose proofs are not even sketched).

Disjoint Chains with Equal Length: The Niederreiter-Rosenbloom-Tsfasman Metric

The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel... more The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel developments.

Unrestricted Generalized Column Distances: A Wider Definition

2019 IEEE International Symposium on Information Theory (ISIT), 2019

In this work we introduce the concept of Unrestricted Generalized Column Distance (UGCD) for conv... more In this work we introduce the concept of Unrestricted Generalized Column Distance (UGCD) for convolutional codes. This is the concept equivalent to the generalized Hamming weights for block codes. We show that the hierarchy of UGCD is strictly increasing and show how to compute it of parity-check matrix in general form. We also provide a way to compute it out of a systematic parity-check matrix.

Mensagens Secretas com Matrizes

Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric spac... more Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(Ak) decreases to 0 if Ak is a sequence of matrices in SL(n, R) with limk→∞ ‖Ak‖ = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained. The sphere packing problem is one of the famous open problems in mathematics. In short, it asks about the densest way a set of equal spheres can be packed in space n-dimensional Euclidean space R, without overlapping one the other. In this context, the density means the proportion between the covered and the uncovered amount of space. It has many variations: one could replace spheres of equal radii by spheres of radii 0 < a ≤ r ≤ b bounded from above and below, replace spheres by a collection of identical (preferably convex) bodies,...

resumos do..., 2019

Estudo do Rendimento do Código SPC Produto Resumo Códigos Single Parity Check (SPC) são [n, k]-có... more Estudo do Rendimento do Código SPC Produto Resumo Códigos Single Parity Check (SPC) são [n, k]-códigos binários lineares, onde k = n-1, cujas palavras sempre possuem peso de Hamming par. Além disso, todo código SPC é MDS, implicando em distância mínima d=2. Códigos SPC produto são construidos a partir do produto tensorial entre códigos SPC. Nesse caso, estudamos apenas o produto entre um código SPC consigo mesmo, resultando nos parâmetros [n², (n-1)², 4]. Sobre um canal binário de apagamentos (BEC), são capazes de corrigir qualquer padrão de até 3 apagamentos e, em alguns casos, corrigem padrões de 4 a 2n-1 erros. Para estudar este problema, utilizou-se abordagens baseadas em análise combinatória, teoria dos grafos bipartidos e dependência linear em matrizes binárias, com intuito de realizar a contagem de padrões corrigíveis para n>8.

Journal of Communication and Information Systems, 2004

MacWilliams' Identity for metrics determined by directed graphs

2016 IEEE Information Theory Workshop (ITW), 2016

Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characteri... more Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characterize the graphs such that every linear code admits a G-canonical decomposition. This decomposition will play an important role in this work, since it will be the main tool to give a sufficient condition for a finite directed graph to satisfy both the MacWilliams Identity and the MacWilliams Extension Property.

A low complexity coding and decoding strategy for the quadratic Gaussian CEO problem

Journal of the Franklin Institute, 2016

Abstract We consider the quadratic Gaussian CEO problem, where the goal is to estimate a measure ... more Abstract We consider the quadratic Gaussian CEO problem, where the goal is to estimate a measure based on several Gaussian noisy observations which must be encoded and sent to a centralized receiver using limited transmission rate. For real applications, besides minimizing the average distortion, given the transmission rate, it is important to take into account memory and processing constraints. Considering these motivations, we present a low complexity coding and decoding strategy, which exploits the correlation between the measurements to reduce the number of bits to be transmitted by refining the output of the quantization stage. The CEO makes an estimate using a decoder based on a process similar to majority voting. We derive explicit expression for the CEO׳s error probability and compare numerical simulations with known achievability results and bounds.

Espaços poset e o problema da distribuição de pesos

Neste trabalho fazemos uma apresentacao dos espacos poset, introduzidos por Brualdi (1995), apres... more Neste trabalho fazemos uma apresentacao dos espacos poset, introduzidos por Brualdi (1995), apresentamos os conceitos necessarios da teoria de conjuntos parcialmente ordenados e da teoria de codigos. Trabalhamos com uma questao de carater amplo e estrutural deste contexto, o problema da determinacao da ordem atraves da distribuicao de pesos. A distribuicao de pesos e essencialmente o conjunto das cardinalidades das esferas metricas e a pergunta que se coloca e em que medida este invariante determina a metrica em questao. Demonstramos que para as classes de codigos, cadeia, anticadeia, coroa e hierarquico, classes importantes no contexto da teoria de codigos, o problema possui uma resposta positiva e justificamos algumas conjecturas que relacionam este problema ao da reconstrucao de grafos Abstract

This work is about two topics in Coding and Information Theory in hyperbolic spaces: (i) Error pr... more This work is about two topics in Coding and Information Theory in hyperbolic spaces: (i) Error probability of signal transmissions in hyperbolic spaces; (ii) Signal constellations with geometric properties feasible for practical applications in environments with hyperbolic behavior. In the first item, it was developed an upper bound for the error probability of signal transmissions in ndimensional hyperbolic space. It was obtained a class of probability density functions for hyperbolic noise analogous to the Gaussian noise in n-dimensional Euclidean space, that we called "hyperbolic Gaussian noise". In this way, the performance comparison between hyperbolic signal constellations perturbed by this type of noise is reasonable for computation. In addition, signal constellations of hyperbolic M-PSK type are analyzed in terms of performance concerning error probability. Related to the second item it was obtained both geometrically and non-geometrically uniform signal constellations on surfaces constructed through quotients of hyperbolic spaces by discrete groups of isometries. The infinite constellations over non-compact surfaces obtained by this process are similar to chrystallographic group lattices in Euclidean space. The finite non-geometrically uniform constellations over compact surfaces are generated by self-intersection of unique knots over the torus of genus g and are cyclic labeled by the group Z n. In addition, an analysis of transmission in channels perturbed by "hyperbolic Gaussian noise", according to the description of the first item is develloped.

Revista Professor de Matemática On line, 2022

Este texto apresenta uma sequência de atividades didáticas para ser realizada com alunos do Ensin... more Este texto apresenta uma sequência de atividades didáticas para ser realizada com alunos do Ensino Básico, com o objetivo de explorar o funcionamento do brinquedo chamado espirógrafo. Os conteúdos matemáticos utilizados e explorados são elementares (proporcionalidade, mínimo múltiplo comum e máximo divisor comum), e a sequência visa propiciar aos alunos uma descoberta orientada, baseada em observações de fenômenos, e concluída com uma demonstração transformacional.

Poset Codes: Partial Orders, Metrics and Coding Theory

SpringerBriefs in Mathematics, 2018

Basic Concepts of Coding Theory

Poset Codes: Partial Orders, Metrics and Coding Theory, 2018

We briefly introduce some very basic facts about the theory of error correcting codes or coding t... more We briefly introduce some very basic facts about the theory of error correcting codes or coding theory. To be more precise, we introduce those basic facts that are related to or dependent on the concept of metric. Many hypothesis are presented in a simplified form or in a restricted context, except for the concept of metric, which is considered in full generality, not really specified at this point, since this is the main goal of these notes: to show the role that different metrics may have in the context of coding theory and to present some interesting questions and bias to the study of finite metric spaces that arouse from coding theory problems and invariants.

2017 IEEE International Symposium on Information Theory (ISIT), 2017

In this work, we adapt the notion of generalized Hamming weight of block codes to introduce the n... more In this work, we adapt the notion of generalized Hamming weight of block codes to introduce the novel concept of generalized column distances for convolutional codes. This can be considered as an extension of the work done in [18] on the generalized Hamming weights for free distance of convolutional codes. We also introduce the concept of Almost-MDP and Near-MDP convolutional code. The problem of constructing convolutional codes with design generalized column distances remains an interesting open problem that requires further research.

Disjoint Chains with Equal Length: The Niederreiter-Rosenbloom-Tsfasman Metric

Poset Codes: Partial Orders, Metrics and Coding Theory, 2018

The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel... more The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel developments.

Special issue on “Many Faces of Distances”

International Transactions in Operational Research, 2014

ArXiv, 2018

In this work we explore the family of metrics determined by SSS-weights, i.e., non-negative funct... more In this work we explore the family of metrics determined by SSS-weights, i.e., non-negative functions over finite fields that respect the support. First, we introduce a conditional sum of weights and classify those which every set of equivalent weights is closed under such sums. Then, we introduce an structure to represent all decision criteria which allows us to characterize the group of linear isometries for SSS-weights sharing the same equivalence class regarding the decoding criterion.

Caracterização de métricas induzidas por posets hierárquicos

Neste trabalho, utilizando a decomposicao canonica para codigos em espacos munidos de metricas de... more Neste trabalho, utilizando a decomposicao canonica para codigos em espacos munidos de metricas definidas por ordens parciais hierarquicas, sao apresentadas formulas explicitas para os principais parâmetros de um codigo linear (distância minima, raios de empacotamento, de cobertura e de Chebyshev) em termo dos correspondentes na metrica de Hamming. Alem disso, e apresentada uma serie de propriedades que caracterizam as metricas definidas por ordens parciais hierarquicas, sendo algumas das caracteristicas originais e, para todas estas propriedades, apresentamos demonstracoes originais, simples e curtas. Abstract

Poset Metrics

SpringerBriefs in Mathematics, 2018

Generalizations, Variations and Perspectives

In this chapter we present different generalizations and variations of the poset metrics. The app... more In this chapter we present different generalizations and variations of the poset metrics. The approach adopted here is very concise. For each of these generalizations we introduce the proper definitions and only explain the concepts needed to understand the statement of the main results (whose proofs are not even sketched).

Disjoint Chains with Equal Length: The Niederreiter-Rosenbloom-Tsfasman Metric

The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel... more The study of codes for the Niederreiter-Rosenbloom-Tsfasman metric has a long history of parallel developments.

Unrestricted Generalized Column Distances: A Wider Definition

2019 IEEE International Symposium on Information Theory (ISIT), 2019

In this work we introduce the concept of Unrestricted Generalized Column Distance (UGCD) for conv... more In this work we introduce the concept of Unrestricted Generalized Column Distance (UGCD) for convolutional codes. This is the concept equivalent to the generalized Hamming weights for block codes. We show that the hierarchy of UGCD is strictly increasing and show how to compute it of parity-check matrix in general form. We also provide a way to compute it out of a systematic parity-check matrix.

Mensagens Secretas com Matrizes

Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric spac... more Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(Ak) decreases to 0 if Ak is a sequence of matrices in SL(n, R) with limk→∞ ‖Ak‖ = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained. The sphere packing problem is one of the famous open problems in mathematics. In short, it asks about the densest way a set of equal spheres can be packed in space n-dimensional Euclidean space R, without overlapping one the other. In this context, the density means the proportion between the covered and the uncovered amount of space. It has many variations: one could replace spheres of equal radii by spheres of radii 0 < a ≤ r ≤ b bounded from above and below, replace spheres by a collection of identical (preferably convex) bodies,...

resumos do..., 2019

Estudo do Rendimento do Código SPC Produto Resumo Códigos Single Parity Check (SPC) são [n, k]-có... more Estudo do Rendimento do Código SPC Produto Resumo Códigos Single Parity Check (SPC) são [n, k]-códigos binários lineares, onde k = n-1, cujas palavras sempre possuem peso de Hamming par. Além disso, todo código SPC é MDS, implicando em distância mínima d=2. Códigos SPC produto são construidos a partir do produto tensorial entre códigos SPC. Nesse caso, estudamos apenas o produto entre um código SPC consigo mesmo, resultando nos parâmetros [n², (n-1)², 4]. Sobre um canal binário de apagamentos (BEC), são capazes de corrigir qualquer padrão de até 3 apagamentos e, em alguns casos, corrigem padrões de 4 a 2n-1 erros. Para estudar este problema, utilizou-se abordagens baseadas em análise combinatória, teoria dos grafos bipartidos e dependência linear em matrizes binárias, com intuito de realizar a contagem de padrões corrigíveis para n>8.

Journal of Communication and Information Systems, 2004

MacWilliams' Identity for metrics determined by directed graphs

2016 IEEE Information Theory Workshop (ITW), 2016

Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characteri... more Considering metrics based on finite directed graph, introduced by Etzion and Firer, we characterize the graphs such that every linear code admits a G-canonical decomposition. This decomposition will play an important role in this work, since it will be the main tool to give a sufficient condition for a finite directed graph to satisfy both the MacWilliams Identity and the MacWilliams Extension Property.

A low complexity coding and decoding strategy for the quadratic Gaussian CEO problem

Journal of the Franklin Institute, 2016

Abstract We consider the quadratic Gaussian CEO problem, where the goal is to estimate a measure ... more Abstract We consider the quadratic Gaussian CEO problem, where the goal is to estimate a measure based on several Gaussian noisy observations which must be encoded and sent to a centralized receiver using limited transmission rate. For real applications, besides minimizing the average distortion, given the transmission rate, it is important to take into account memory and processing constraints. Considering these motivations, we present a low complexity coding and decoding strategy, which exploits the correlation between the measurements to reduce the number of bits to be transmitted by refining the output of the quantization stage. The CEO makes an estimate using a decoder based on a process similar to majority voting. We derive explicit expression for the CEO׳s error probability and compare numerical simulations with known achievability results and bounds.