Induced subarrays of Latin squares without repeated symbols

We show that for any Latin square L of order 2m, we can partition the rows and columns of L into pairs so that at most (m+3)/2 of the 2x2 subarrays induced contain a repeated symbol. We conjecture that any Latin square of order 2m (where m ≥ 2, with exactly five transposition class exceptions of order 6) has such a partition so that every 2x2 subarray induced contains no repeated symbol. We verify this conjecture by computer when m ≤ 4

Balanced diagonals in frequency squares

We say that a diagonal in an array is λ-balanced if each entry occurs λ times. Let L be a frequency square of type F (n; λ); that is, an n ✕ n array in which each entry from {1, 2, …, m=n / λ } occurs λ times per row and λ times per column. We show that if m≤3 , L contains a λ -balanced diagonal, with only one exception up to equivalence when m=2. We give partial results for m≥4 and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array

The linear system for Sudoku and a fractional completion threshold

We study a system of linear equations associated with Sudoku latin squares. The coefficient matrix $M$ of the normal system has various symmetries arising from Sudoku. From this, we find the eigenvalues and eigenvectors of $M$, and compute a generalized inverse. Then, using linear perturbation methods, we obtain a fractional completion guarantee for sufficiently large and sparse rectangular-box Sudoku puzzles

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Design of terahertz transceiver schemes for ultrahigh-speed wireless communications

Future ultra-high-speed wireless communication systems face difficult challenges due to the fundamental limitations of current technologies operating at microwave frequencies. Supporting high transmission rates will require the use of more spectral resources that are only available at higher frequencies. Within this context, terahertz (THz) communications have been attracting more and more attention, being considered by the research community as one of the most promising research fields on the topic due to the availability of extensive unused bandwidth segments. However, its widespread use is not yet possible due to some obstacles, such as the high propagation losses that occur in this band and the difficulty in designing devices that can effectively perform both transmission and detection tasks. The purpose of this dissertation is to contribute for the solution of both of the aforementioned problems and to propose novel THz transceiver schemes for ultra-high-speed wireless communications. Three main research areas were addressed: device modelling for the THz; index modulation (IM) based schemes for Beyond 5G (B5G) networks and hybrid precoding designs for THz ultra massive (UM) – multiple input multiple output (MIMO) systems. The main contributions of this work include the creation of a new design for a reconfigurable THz filter; the proposal of a precoded generalized spatial modulation scheme for downlink MIMO transmissions in B5G networks; the creation of a low-complexity hybrid design algorithm with a near fully-digital performance for multiuser (MU) mmWave/THz ultra massive MIMO systems that can incorporate different analog architectures; and the system-level assessment of cloud radio access network (C-RAN) deployments based on low-complexity hybrid precoding designs for massive MIMO downlink transmissions in B5G networks. The first contribution is especially suited for the implementation of reconfigurable THz filters and optical modulators, since it is based on a simple design, which transits from situations in which it presents a full transparency to situations where it achieves full opacity. Moreover, this approach can also be used for the implementation of simultaneously transmitting and reflecting (STAR) reconfigurable intelligent surfaces (RIS) which are important for enabling flexible system designs in RIS-assisted networks. The second contribution showed that the implementation of precoding schemes based on generalised spatial modulations is a solution with a considerable potential for future B5G systems, since it can provide larger throughputs when compared to conventional MU-MIMO schemes with identical spectral efficiencies.The last two contributions showed that through the proposed hybrid design algorithm it becomes possible to replace a fully digital precoder/combiner by a fully-connected or even by a partially-connected architecture (array of subarrays and dynamic array of subarrays), while achieving good tradeoffs between spectral efficiency, power consumption and implementation complexity. These proposals are particularly relevant for the support of UM-MIMO in severely hardware constrained THz systems. Moreover, the capability of achieving significant improvements in terms of throughput performance and coverage over typical cellular networks, when considering hybrid precoding‐based C-RAN deployments in two indoor office scenarios at the THz band, was demonstrated.Os futuros sistemas de comunicação sem fios de velocidade ultra-elevada enfrentam desafios difíceis devido às limitações fundamentais das tecnologias atuais que funcionam a frequências de microondas. O suporte de taxas de transmissão altas exigirá a utilização de mais recursos espectrais que só estão disponíveis em frequências mais elevadas. A banda Terahertz (THz) é uma das soluções mais promissoras devido às suas enormes larguras de banda disponíveis no espectro eletromagnético. No entanto, a sua utilização generalizada ainda não é possível devido a alguns obstáculos, tais como as elevadas perdas de propagação que se verificam nesta banda e a dificuldade em conceber dispositivos que possam desempenhar eficazmente as tarefas de transmissão e deteção. O objetivo desta tese de doutoramento, é contribuir para ambos os problemas mencionados anteriormente e propor novos esquemas de transcetores THz para comunicações sem fios de velocidade ultra-elevada. Três grandes áreas de investigação foram endereçadas, contribuindo individualmente para um todo: a modelação do dispositivo para o THz; esquemas baseados em modulações de índice (IM) para redes pós-5G (B5G) e desenhos de pré-codificadores híbridos para sistemas THz MIMO ultra-massivos. As principais contribuições deste trabalho incluem a criação de um novo design para um filtro THz reconfigurável; a proposta de uma nova tipologia de modulação espacial generalizada pré-codificada para transmissões MIMO de ligação descendente para redes B5G; a criação de um algoritmo de design híbrido de baixa complexidade com desempenho quase totalmente digital para sistemas MIMO multi-utilizador (MU) mmWave/THz ultra massivos que podem incorporar diferentes arquiteturas analógicas e a avaliação das implementações da rede de acesso de rádio na nuvem (C-RAN) com base em designs de pré-codificação híbridos de baixa complexidade para transmissões MIMO de ligação descendente massivas em redes B5G. A primeira contribuição é especialmente adequada para a implementação de filtros THz reconfiguráveis e moduladores óticos, uma vez que se baseia numa concepção mais simples, que transita de situações em que apresenta uma transparência total para situações em que atinge uma opacidade total. Para além disso, esta abordagem também pode ser utilizada para a implementação de superfícies inteligentes reconfiguráveis (RIS) de transmissão e reflexão simultânea (STAR). A segunda contribuição mostrou que a implementação de esquemas de pré-codificação baseados em modulações espaciais generalizadas é uma solução com um potencial considerável para futuros sistemas B5G, uma vez que permite alcançar maiores ganhos em termos de débito binário quando comparado com esquemas convencionais MU-MIMO com eficiências espectrais idênticas. As duas últimas contribuições mostraram que através do algoritmo proposto torna-se possível substituir a utilização de uma arquitectura totalmente digital por uma arquitetura totalmente conectada ou mesmo por uma arquitetura parcialmente conectada (arrays de subarrays e arrays dinâmicos de subarrays), conseguindo-se bons tradeoffs entre eficiência espectral, consumo de energia e complexidade de implementação. Estas propostas são particularmente relevantes para dar suporte a sistemas THz UM-MIMO com restrições severas ao nível de hardware. Demonstrou-se também a capacidade de se alcançar melhorias significativas em termos de débito binário e cobertura em relação a redes celulares típicas, considerando dois cenários na banda THz

Synchronizing permutation groups and graph endomorphisms

The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form , and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8)

Ionosphere Monitoring with Remote Sensing

This book focuses on the characterization of the physical properties of the Earth’s ionosphere, contributing to unveiling the nature of several processes responsible for a plethora of space weather-related phenomena taking place in a wide range of spatial and temporal scales. This is made possible by the exploitation of a huge amount of high-quality data derived from both remote sensing and in situ facilities such as ionosondes, radars, satellites and Global Navigation Satellite Systems receivers

Towards a more efficient spectrum usage: spectrum sensing and cognitive radio techniques

The traditional approach of dealing with spectrum management in wireless communications has been through the definition on a license user granted exclusive exploitation rights for a specific frequency.Peer ReviewedPostprint (published version