A trifferent code of length n is a subset of {0, 1, 2}ⁿ such that for any three distinct elements in it, there is a coordinate in which they all differ pairwise. The quantity T(n) denotes the largest size of such a code of length n. A motivation to study this problem comes from an application in information theory, where T(n) is related to the zero-error capacity of the (3/2)-channel, defined by Elias in 1988. This problem has seen renewed interest due to recently established connections between linear trifferent codes, strong blocking sets in projective geometry and minimal codes in coding theory.
In this thesis, we examine the most recent breakthrough in the upper bound of T(n) by Bhandari and Khetan, coming from r-bounded trifferent codes, which are trifferent codes with each codeword having exactly r many 2s. The quantity T_b(n,r) denotes the largest size of r-bounded trifferent codes of length n.
We generalize previous results by Bhandari and Khetan to give the upper bound T_b(n, r) ≤ c × n^r−2/5 for all r ≥ 3. We also build upon ideas given by Bishnoi and Kovács and prove the lower bound T_b(n, r) ≥ n^{⌈r/2⌉−o(1)} for all r ≥ 3 using special existing hypergraph constructions. In order to improve the lower bound for the quantity T_b(n, 2), we use a SAT solver to compute small-sized 2-bounded trifferent codes. With the help of these computations, we come up with two new constructions for 2-bounded trifferent codes which improve the prevailing (trivially obtained) lower bound of T_b(n, 2) ≥ 2n – 2 to T_b(n, 2) ≥ 2n (from Construction 1, in joint work with Jozefien D’haeseleer) and an even better
bound of T_b(n, 2) ≥ (20/9)n – O(1) (from Construction 2). ... Read more

This thesis addresses the t-avoiding-set problem on the complex n-dimensional unit sphere, which asks for the maximal surface measure of a set where no pair of points has an inner product equal to t. By first interpreting the t-avoiding-set problem as an independence-number problem, we use a formulation for the Lovasz theta number to find upper bounds for the maximum measure. In order to adapt the formulation for use in standard optimization solvers, we construct real-valued disk polynomials and use them as a basis for solutions. We further improve the upper bounds by extending the formulation for the Lovasz theta number with a set of constraints derived using the Boolean Quadric Polytope (BQP). In this thesis, we find an optimal construction for eiϕ-avoiding sets and analyze the behavior of the upper bounds for t-avoiding sets. Given that the 0-avoiding set problem corresponds to Witsenhausen’s problem on the complex sphere, we investigate this problem and its upper bounds in depth. ... Read more

A nowhere-zero flow on a directed graph is a function that assigns a value to each edge such that each vertex has equal in- and outflow. This concept is generalized to d-dimensional nowhere-zero flows. The two-dimensional flow number of a graph is the smallest number r such that the graph has a two-dimensional nowhere-zero flow using only vectors with lengts between 1 and r-1.

In this paper, two-dimensional flow numbers are determined for several graphs. First, these are determined for all complete multipartite graphs. Second, flow triangulations are researched, and a flow triangulation is found for the Wagner graph. Furthermore, this research addresses the question whether a nice flow triangulation exists for all bipartite cubic graphs. Furthermore, optimization models are used to approximate two-dimensional flow numbers of certain graphs, including snarks.
... Read more

Efficient staff scheduling is crucial for smooth baggage handling at Schiphol Airport. Each day, KLM must allocate hundreds of employees to hundreds of time-sensitive loading and unloading tasks. Even small disruptions, such as flight delays or staff shortages, can quickly lead to infeasible schedules and unhandled flights. This thesis develops an optimization framework to provide more insight into the robustness of KLM’s day-ahead baggage handling schedule.

The problem is modeled as a Vehicle Routing Problem with Time Windows (VRPTW), extended mode selection, travel times, and mandatory breaks. To handle the model’s complexity, a Fix-and-Optimize approach is applied, iteratively optimizing subsets of the problem while keeping others fixed. Two extensions are introduced to improve task coverage: allowing staff to assist across work areas (cross-zone assignments) and slightly delaying flights (postponements) when necessary.

Scenario analyses evaluate how disruptions, flight delays and staff shortages, affect task coverage. Results based on real KLM data show that the Base Model already increases the number of planned tasks compared to the current planning tool when comparing optimization for a full shift. Allowing postponements further improves coverage, often enabling full task completion. Under disruptions, the model identifies critical time periods and areas where additional flexibility or actions are required.

This research combines mathematical modeling and analysis to better understand the structure and resilience of KLM’s daily planning. The proposed approach highlights where schedules are vulnerable and demonstrates how optimization can support planners in preparing for operational uncertainty. ... Read more

This research aims to optimize a VRPTW that incorporates the driver satisfaction factors ‘region consistency’ and ‘workload preference’ while not increasing routing costs too much. The developed measures were optimized for using the ‘Random Allocation’, ‘Driver Assignment’ and ‘Integrated Approach’ methods for various weightings in a multi-objective setting. Driver satisfaction was explicitly optimized in the state-of-the-art VRPTW solver called PyVRP developed by ORTEC. The integrated approach outperformed both the driver assignment and random allocation methods on small and medium sized instances, whereas Driver Assignment performed best on large instances. ... Read more

We solve steps of the Lasserre hierarchy for various topological packing graphs. New techniques were necessary to do so and this thesis focuses on the calculation of the zonal Stiefel harmonics. First, we develop an appropriate harmonic analytical framework. To use this, the subspaces of representations of the orthogonal group, which are invariant under a subgroup, have to be determined. Then, the matrix coefficients of these invariants have to be calculated in practice. We give two methods to do so. For the equiangular lines problem with a fixed angle this leads to new bounds. For the kissing number problem in dimension four, a sharp optimal solution is obtained. Via complementary slackness, this leads to the new result that the D4 root system is the unique optimal kissing configuration in dimension four. ... Read more

In this thesis, we consider extremal problems in discrete geometry, and in particular, the Lasserre hierarchies for such problems. We give a unifying framework that encompasses the known hierarchies, and lay the foundation to use the second level of such hierarchies for problems on the sphere in practice. We perform explicit harmonic analysis and use polynomial optimization techniques to reduce the problems to a finite-dimensional semidefinite program. We introduce a specialized semidefinite programming solver that uses the structure of the problems, allowing us to use polynomials of significantly higher degree than previously possible, and a much faster rounding procedure to obtain exact optimal solutions to the semidefinite programs.
We use this to prove that the D₄ root system is the unique optimal solution to the kissing number problem in dimension 4, and is an optimal spherical code. We also prove there are exactly two optimal spherical codes with 12 points in dimension 4. Furthermore, we show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes, as well as certain Kerdock spherical codes. We give numerical evidence that the second level of the Lasserre hierarchy for minimizing harmonic energy is sharp for several infinite families of configurations.
We also investigate the strength of the hierarchy for the polarization problem. Finally, we consider triple and quadruple correlation bounds in analytic number theory, which gives new bounds on the fraction of double and triple zeros of the Riemann ζ-function and related functions. ... Read more

The objective of the cap set problem is finding the maximum size of a d-cap: a subset of  𝔽₃^d not containing three elements in line. This thesis aims to give a comprehensive overview of constructions for the cap set problem, with a focus on improvements of the asymptotic lower bound on the size of caps that have already been made.

Finding an asymptotic lower bound on the size of caps boils down to finding a cap C in a dimension d such that its solidity, given by |D|^1/d, is as large as possible. We start with studying caps in low dimensions, of which the maximum sizes are exactly known. Then to further improve the asymptotic lower bound we turn to caps in higher dimensions. Here, the art lies in carefully combining large caps in low dimensions to construct large caps in higher dimensions by taking products. One construction that allows us to do this is the extended product construction, which extends extendable collections of caps with admissible sets.

This thesis explains the extended product construction and gives an overview of how it has been used and expanded to repeatedly increase the asymptotic lower bound. As the literature sometimes lacks detail, this thesis adds to the literature by incorporating examples, explicit constructions of (recursively) admissible sets, and experiments with the extended product construction.

In Chapter 6, we prove the existence of recursively admissible sets of constant weight 2 and 3 for any dimension k by giving explicit constructions and proving that the resulting sets satisfy all necessary conditions. Moreover, we classify all admissible sets in dimensions 2 and 3 and all extendable collections in dimensions 1, 2, and 3. Then, we use these to deduce that the extended product construction is less effective in low dimensions by showing that the largest possible caps we can construct this way in dimensions 4, 6, and 8 are never as large as caps constructed by taking direct products of maximum caps. ... Read more

Consider an arbitrary finite grid in some field. How many hyperplanes are required so that every point is contained in at least k hyperplanes, except for one point that is not allowed to be contained in any hyperplane? To solve this so-called hyperplane grid covering problem, the polynomial method has proven to be extremely useful in finding bounds on the minimum number of hyperplanes required. This has given rise to a second problem: the polynomial grid covering problem. This problem considers the minimum degree of a polynomial such that every grid point is a root with multiplicity k, except for one point where the polynomial does not vanish. We study these two related problems for multiple grids: the hypercube, the vector space over the binary field and grids in the Cartesian plane. Since polynomial covers in the latter have not been studied before, we provide algorithms and techniques to study these covers. We also explore the link between grid coverings and two results from algebraic geometry: the Footprint Bound and the Cayley-Bacharach Theorems. ... Read more

Matrix multiplication with the standard algorithm has an algebraic complexity of O(n^3) for n x n matrices, but in 1969 Strassen found another algorithm for multiplying 2 x 2 matrices with which he showed that matrix multiplication can be done with a complexity of O(n^2.81) by applying his algorithm recursively for large matrices. We present a historical overview of the best known bounds on the matrix multiplication exponent, with the current best known bound of 2.371866 and methods, used to find new algorithms for other matrix multiplications, such as alternating least squares (ALS) and SAT solving. After this we present a novel method with which we found a rank 23 decomposition of the <3,3,3> matrix multiplication tensor that may be inequivalent to known decompositions. We also found decompositions for <2,2,2>, <2,2,3>, <2,2,4>, <2,3,3> and <2,2,5> that provide the same bounds as earlier found decompositions. Our method found many decompositions for the smaller tensors but only one for the larger due to time limits. The method is based on the alternating least squares method (ALS) with the modification that we 'push' the coefficients towards integer (or rational) numbers. After a number of iterations and rounding the result this sometimes yields exact decompositions so that we can bound the matrix multiplication exponent. ... Read more

A set of lines passing through the origin in Euclidean space is called equiangular if the angle between any two lines is the same. The question of finding the maximum number of such lines, N(d) in any dimension d is an extensively studied problem. Closely related, is the problem of finding the maximum number of lines, N_α(d), such that the common angle between the lines is arccosα. In recent years, many progress has been made on this problem. We review some of these breakthrough results and the techniques they use to approach this problem. The first main result is a linear upper bound on N_α(d) which is found using a completely novel approach with respect to techniques used in previous works. Another main result that we discuss solves the problem of finding N_α(d) for high enough dimensions. Some classic results from some of the first studies on equiangular lines are also discussed. Finally, some suggestions are given for possible further research. ... Read more

The increasing popularity of e-commerce has led to a greater emphasis on improving parcel delivery processes. Among the various stages of the delivery process, packing parcels into delivery vans affects the delivery time. The efficiency of delivery is optimized when each parcel is conveniently accessible upon arrival, thereby minimizing the requirement for additional repacking time. Therefore, streamlining parcel packing within delivery vehicles is a crucial aspect of improving overall delivery times in e-commerce. This thesis focuses on developing a packing solution that conforms to the the Last-In-First-Out (LIFO) principle, which is defined as ensuring that a parcel can be accessed by the delivery driver as soon as they reach the destination of the parcel. This is described as the 3D-Bin Packing Problem with Loading Constraints (3L-BPP). To solve this problem, a formulation of a Mixed Integer Linear Program (MILP) has been developed. To improve the speed and accuracy of the solution, a novel placement heuristic has been created. This heuristic is derived from the established Distance to the Front-Top-Right Corner (DFTRC)-2 method and is designed to generate an initial solution.

Both the MILP and the heuristic are tested on a data set of 789 distinct rides, provided by a Dutch postal company. The results demonstrate that (a modified version of) the heuristic successfully generated an initial solution for all rides in the dataset, with 98.9% being found within 3 seconds, and for the remaining 1.1%, the inclusion of a Genetic Algorithm led to a solution being found within 90 seconds. By using the heuristic to establish an initial solution that is then refined through optimization techniques for the MILP, the findings indicate that this approach yields the best outcomes in minimizing the number of incorrectly positioned parcels. ... Read more

This dissertation explores three interconnected topics at the interface of combinatorics, algebra, and geometry.

In Part I, we study the divisorial gonality of graphs, a graph parameter inspired by algebraic geometry. Introduced by Baker around 2007, divisorial gonality links chip-firing games on graphs to classical algebraic geometry, including the combinatorial Riemann–Roch theorem and Brill–Noether theory. We contribute in two ways. First, we provide a constructive proof that treewidth is a lower bound for divisorial gonality, presenting a polynomial-time algorithm that transforms a positive rank effective divisor into a tree decomposition of bounded width. This bridges gonality with structural graph theory and enables dynamic programming techniques for graphs of bounded gonality. Second, we examine the Brill–Noether conjecture for graphs and disprove Baker’s subdivision conjecture by constructing graphs whose gonality exceeds that of the associated metric graphs. While the Brill–Noether existence conjecture remains unresolved, our results clarify structural limitations in approaching its proof.

Part II investigates applications of the slice rank polynomial method to affine configurations over finite fields. Extending the method initially developed to solve the cap set problem, we analyze subsets of
Fqn
F
q
n
​

avoiding non-trivial solutions to systems of balanced linear equations. For certain classes of systems, we demonstrate the existence of solutions with pairwise distinct or maximally affinely independent elements, generalizing previous results by Mimura and Tokushige. These findings contribute to understanding the limitations and potential of slice rank techniques for broader combinatorial problems.

In Part III, we study tensor products of convex cones, a topic relevant across functional analysis, operator theory, approximation theory, and theoretical physics. Existing literature mainly addresses Archimedean lattice cones or finite-dimensional proper generating cones, leaving many cones unexamined. We extend known results to general cones and present several new contributions: we establish mapping properties for projective/injective cones analogous to norms, characterize their lineality spaces and extremal rays, show containment of projective/injective tensor products in proper cones, and demonstrate that tensoring symmetric convex sets preserves proper faces. For closed cones in finite-dimensional spaces, we show that the projective cone is closed and almost always strictly contained in the injective cone, partially recovering Barker’s conjecture prior to its independent full proof by Aubrun et al. Our work unifies tensor product properties across a wide class of convex cones and provides new tools for both theoretical and applied analysis.

Together, these three parts demonstrate the deep interplay between combinatorics, algebra, and geometry, advancing understanding in graph theory, finite field combinatorics, and convex analysis. The results provide constructive algorithms, generalized theoretical frameworks, and novel counterexamples, opening directions for further research in structural graph theory, algebraic combinatorics, and tensor analysis. ... Read more

A code C is defined to be a set of S words, where a word is a sequence of n entries. We call S the size and n the length of the code. The entries of the code can have k different values, {0, .., (k − 1)}. Define a perfect k-hash code (PHC) as a code with the property that any collection of v words in the code is different at at least one index. PHC’s are useful mathematical objects within different fields of theoretical computer science and coding theory. This thesis will focus on one typical kind of PHC, a trifferent code. Such a trifferent code is defined as a PHC where k = v = 3. This means that any collection of three words in the code has to differ at some index. We now define T (n) to be the largest possible size of a trifferent code given that it has length n. The question that arises is, what is the value of T (n)? It turns out that determining the exact value is complicated, unless n is really small. Therefore, we try to understand the asymptotic nature of the function T (n). This is what we call the trifference problem. This paper will cover and prove the best known asymptotic upper and lower bound on T (n). Moreover, it will explain and include a Python code that can be used to show and prove all values of T (n) for n ∈ {1, .., 6}. Lastly, two different ways to explicitly construct trifferent codes for any value of n will be given and compared. ... Read more

This thesis focuses on a problem formulated by Claude Shannon named the Shannon capacity. This problem is about information rate per time unit over a noisy channel. The noisy channel is here represented by a graph. We specifically focus on a class of circulant graphs that are denoted by C(n,k) with vertex set z/nz, where all vertices are connected with the k-1 vertices before and after it. We will discuss upper bounds that were found for the Shannon capacity and how C(n,k) behaves with these upper bounds. After that we will focus on multiple ways to calculate lower bounds for the Shannon capacity of $C(n,k)$. For these three search methods will be used. These are exhaustive searching for optimal values, optimal ways to make packagings and solutions created by using a special form. As last the answers will be discussed by combining the upper and lower bounds for C(n,k). From this conclusions are drawn after which some possibilities will be given for further research.  ... Read more

Representation theory is a branch in mathematics that studies group homomorphisms between a group and the automorphism group of a vector space. A special representation that every group has is the regular representation. This representation permutes all elements of the group in a vector space which dimension is equal to the order of the group.
Within this vector space there are group invariant subspaces. There are several methods to finding representation invariant subspaces of this vector space.

This thesis aims to do two things: First of all, this thesis aims to give the reader an introduction to representation theory, presenting various key concepts, definitions and theorems. Moreover, ways to construct character tables are presented along with multiple worked out examples. Second, the regular representations of D4 and Q8 are decomposed into representation invariant subspaces of C8. To this end, two methods were used.

The first method (A), proposed by dr. Jeroen Spandaw, works out all the possible decompositions of the vector space a regular representation acts on. This is quite a laborious process in which change of basis matrices, expressed in several parameters, will have to be made for all generators of the considered group. The second methods (B), proposed by dr. Paul Visser, is known as the grand orthogonalization method and makes use of an extended version of the character table of the considered group.

Both methods are perfectly fine to make a desired decomposition. However, method A
takes a lot more computational effort than method B to come up with the desired result. The benefit of using method A over method B is that method A considers all possible decompositions, whereas method B only considers one of the infinitely many that are possible. ... Read more

In 1961 Kasteleyn solved the dimer problem. With the use of Pfaffans he managed to find a formula to enumerate the number of perfect matchings on a lattice graph. In this thesis we take another look at the methods Kasteleyn used. Besides that, we proof that for an m × n lattice graph on the torus, where m and n are even, there does not exist a Pfaffan orientation. Instead, we prove that for an m × 2 and 2 × n lattice graph on the torus, where m and n are even, there does exist a Pfaffan orientation. For the m × n lattice graph on the torus, where m is even and n odd we present an orientation together with an algorithm with which we can simplify cycles. With this algorithm we prove that our orientation is Pfaffan. We will now describe the Pfaffan orientation. All horizontal edges are aimed to the right, all vertical edges switch between all going down for the first column, then up for the second column and so on. The edges that are on the border all have orientation opposite of the ongoing orientation. ... Read more

Decision-tree evaluation is a widely-used classification approach known for its simplicity and effectiveness. Decision-tree models are shown to be helpful in classifying instances of fraud, malware, or diseases and can be used to make dynamic, flexible access decisions within an access-control system. These applications often require sensitive data of more than one party, like financial or health-related records. It is important to keep this data private, especially when the decision-tree evaluation is done in a collaborative manner where more than one party provides sensitive input data. Current privacy-preserving solutions only consider scenarios in which input data originates from a single source. However, collaboration for decision-tree evaluation tasks is needed more and more since these collaborations often bear fruit. Therefore, in this work we address Private Decision-Tree Evaluation in a collaborative setting. We assume one entity, called the server, that holds the decision tree and multiple users that provide private data on which the decision tree is evaluated. The focus of our research lies on solutions that make use of homomorphic encryption. We give ten different protocols that each take place in a different setting; either the holder of the decision tree receives the evaluation result or the users that provide the input. The protocols use Multi-Key Fully Homomorphic Encryption (FHE) or normal FHE with a Semi-Trusted Third Party (STTP). Additionally, we introduce a novel key-switching method within two of the STTP protocols such that the dependency on the STTP is greatly reduced. All protocols are proven to be secure in the semi-honest model and compared in terms of run-time complexity and communication costs. Due to the high computational overhead for the Multi-Key FHE schemes, the protocols that make use of these schemes are not yet feasible. Therefore, the protocols that use an STTP are the most promising. All protocols take no more than 4 communication rounds. Assuming that the implementation can be parallelized and given an input bit length of 4, the decision-tree evaluation in our protocols takes in the worst case between 60 and 160 hours, executed on an Intel Core Processor at 2.4 GHz with 16 GB memory. ... Read more

This thesis investigates two types of classical capacities of both classical and quantum channels, giving rise to four different settings. The first type of classical capacity investigated is the ordinary capacity of a channel to transmit classical information with a probability of error which becomes arbitrarily small as the channel is used arbitrarily many times. The second type of classical capacity investigated is the capacity of a channel to transmit information with zero probability of error, called the zero-error classical capacity. The first setting which is studied is the ordinary capacity of a classical channel. The noisy channel coding theorem is proven in two different ways: one using the Markov inequality and the Law of Large Numbers and one using typical sets. The additivity of this capacity is also discussed. The second setting is the zero-error capacity of classical channel. Lower and upper bounds on this capacity are proven, and its superadditivity is discussed. The third setting is the ordinary classical capacity of a quantum channel. The Holevo-Schumacher-Westmoreland theorem is proven using typical subspaces and the packing lemma, and the superadditivity of the Holevo information is discussed in terms of entanglement at the encoder. The fourth and last setting investigated is that of the zero-error classical capacity of a quantum channel. It is shown that this capacity can be achieved using only pure input states and that this capacity never exceeds the ordinary classical capacity. Moreover, a detailed investigation of superactivation of the zero-error classical capacity is presented. A topic for further research would be an exposition of the analogous concepts in the case of the quantum capacity of quantum channels. Another topic for further research would be an explicit construction of two quantum channels whose zero-error classical capacity is superactivated. ... Read more

Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2. ... Read more