Bounds on codes over an alphabet of five elements

AbstractWe consider the problem of finding bounds and exact values of A5(n,d) — the maximum size of a code of length n and minimum distance d over an alphabet of 5 elements. Using a wide variety of constructions and methods, a table of bounds on A5(n,d) for n⩽11 is obtained

The hippocampal formation from a machine learning perspective

Nos dias de hoje, existem diversos tipos de sensores que conseguem captar uma grande quantidade de dados em curtos espaços de tempo. Em muitas situações, as informações obtidas pelos diferentes sensores traduzem fenómenos específicos, através de dados obtidos em diferentes formatos. Nesses casos, torna-se difícil saber quais as relações entre os dados e/ou identificar se os diferentes dados traduzem uma certa condição. Neste contexto, torna-se relevante desenvolver sistemas que tenham capacidade de analisar grandes quantidades de dados num menor tempo possível, produzindo informação válida a partir da informação recolhida. O cérebro dos animais é um órgão biológico capaz de fazer algo semelhante com a informação obtida pelos sentidos, que traduzem fenómenos específicos. Dentro do cérebro, existe um elemento chamado Hipocampo, que se encontra situado na área do lóbulo temporal. A sua função principal consiste em analisar os elementos previamente codificados pelo Entorhinal Cortex, dando origem à formação de novas memórias. Sendo o Hipocampo um órgão que foi sofrendo evoluções ao longo do tempos, é importante perceber qual é o seu funcionamento e, se possível, tentar encontrar modelos computacionais que traduzam o seu mecanismo. Desde a remoção do Hipocampo num paciente que sofria de convulsões, ficou claro que, sem esse elemento, não seria possível memorizar lugares ou eventos ocorridos num determinado espaço de tempo. Essa funcionalidade é obtida através de um conjunto específico de células chamadas de Grid Cells, que estão situadas na área do Entorhinal Cortex, mas também das Place Cells, Head Direction Cells e Boundary Vector Cells. Neste âmbito, o principal objetivo desta Dissertação consiste em descrever os principais mecanismos biológicos localizados no Hipocampo e definir modelos computacionais que consigam simular as funções mais críticas de ambos os Hipocampos e da área do Entorhinal Cortex.Nowadays, sensor devices are able to generate huge amounts of data in short periods of time. In many situations, that data, collected by many different sensors, translates a specific phenomenon, but is presented in very different types and formats. In these cases, it is hard to determine how these distinct types of data are related to each other or translate a certain condition. In this context, it would be of great importance to develop a system capable of analysing such data in the smallest amount time to produce valid information. The brain is a biological organ capable of such decisions. Inside the brain, there is an element called Hippocampus, that is situated in the Temporal Lobe area. Its main function is to analyse the sensorial data encoded by the Entorhinal Cortex to create new memories. Since the Hippocampus has evolved for thousands of years to perform these tasks, it is of high importance to try to understand its functioning and to model it, i.e. to define a set of computer algorithms that approximates it. Since the removal of the Hippocampus from a patient suffering from seizures, the scientific community believes that the Hippocampus is crucial for memory formation and for spatial navigation. Without it, it wouldn’t be possible to memorize places and events that happened in a specific time or place. Such functionality is achieved with the help of set of cells called Grid Cells, present in the Entorhinal Cortex area, but also with Place Cells, Head Direction Cells and Boundary Vector Cells. The combined information analysed by those cells allows the unique identification of places or events. The main objective of the work developed in this Thesis consists in describing the biological mechanisms present in the Hippocampus area and to define potential computer models that allow the simulation of all or the most critical functions of both the Hippocampus and the Entorhinal Cortex areas

On Linear Codes over F2 x F2

A code of length n and size M consist of a set of M vectors of n components. The components being taken from some alphabet set S. So a code C is a set of n-tuples subset of Sn. If S has a ring structure then C is called a linear code over S if it is an S-module. To every linear code C there corresponds its dual C⊥, if C C⊥, then C is called self-orthogonal. If C = C⊥ then C is called self-dual. In this thesis we will study linear and self-dual codes over the rings of four alphabets and in more details over the ring F2 x F2, this ring is isomorphic to the ring F2 + vF2 where v2 = v and F2 = {0; 1}. We would also study linear and self-dual codes for other rings in the form Fp + vFp for different primes p. Also we will construct simplex code over the ring F2 + vF2≃ F2 x F2

Designing New Structures of Magnetic Materials: Cases of Metal Borides and Metal Chalcogenides

We are in the middle of a changing paradigm in solid-state chemistry: shifting away from analyzing the structures of crystalline compounds to understand the structure-property relations towards the targeted synthesis of new materials with properties desirable for specific applications – known as the materials-by-design strategy.This approach requires control over the obtained products and their structures, which is still an enormously challenging task, especially for intermetallics. Intermetallics are non-molecular crystalline solids based on two or more metals with a stoichiometry that is not continuously variable and for which the crystal structure differs from the structure of the involved elements. Metal-rich borides are a subgroup of intermetallics with huge structural variety and highly interesting properties. Many metal borides are hard (or even superhard), have extremely high melting points, and exhibit fascinating magnetic properties (if they contain magnetically active elements).This work primarily focuses on metal-rich borides and their structural relations but it includes some chalcogenides as well. A ternary niobium boride, Nb1-xOs1+xB with a new crystal structure was discovered, presented itself to be a missing link. Its discovery led to the identification of a new class of borides containing over a dozen different crystal structures. The structure-building principles were explored, and it was found that all of these crystal structures can be described using the same primary and secondary building blocks. The structure of Ti5-xFe1-yOs6+x+yB6, another boride in a new crystal structure with interesting magnetic properties due to 1-dimensional Fe-chains, can be explained in the same conceptual framework. Moreover, for the first time, hitherto unknown crystal structures with exciting magnetic properties can be predicted by applying these structure-building principles. The new TiFe1-xOs2+xB2, which is predicted using such principles, is one of the few ternary metal-borides without rare-earth elements that is ferromagnetic above room temperature.The structure-building principles with their predictive power can serve as guidelines for obtaining metal borides with specific properties. Therefore, they are an important step towards materials-by-design. Moreover, they could aid in understanding how and why observed structures form while others do not, a crucial step to control and ultimately manipulate the formation of these compounds in the future

Non–existence of some 4–dimensional Griesmer codes over finite fields

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$, $2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and $q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$ and that $n_q(4,d) \ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and $2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists

Synchronous code-division multiplex systems

The investigation is concerned with various synchronous multiplexing and demultiplexing processes suitable for use with serial baseband data-transmission systems. The multiplexed signals are transmitted in orthogonal groups over a channel which introduces additive white Gaussian noise but no signal distortion. Techniques are considered for increasing both the capacity and tolerance to additive noise, when the number of multiplexed signals may vary with time, and may exceed the maximum number of orthogonal multiplexed signals. [Continues.