This thesis uses algebraic and combinatorial methods to study subsets of the Desarguesian plane IIq = PG(2, q). Emphasis, in particular, is given to complete (k, n)-arcs and plane projective curves. Known Diophantine equations for subsets of PG(2, q), no more than n of which are collinear, have been applied to k-arcs of arbitrary degree. This yields a new lower bound for complete (k, n)-arcs in PG(2, q) and is a generalization of a classical result of Barlotti. The bound is one of few known results for complete arcs of arbitrary degree and establishes new restrictions upon the parameters of associated projective codes. New results governing the relationship between (k, 3)-arcs and blocking sets are also provided. Here, a sufficient condition ensuring that a blocking set is induced by a complete (k, 3)-arc in the dual plane q is established and shown to complement existing knowledge of relationships between k-arcs and blocking sets. Combinatorial techniques analyzing (k, 3)-arcs in suitable planes are then introduced. Utilizing the numeric properties of non-singular cubic curves, plane (k, 3)-arcs satisfying prescribed incidence conditions are shown not to attain existing upper bounds. The relative sizes of (k, 3)-arcs and non-singular cubic curves are also considered. It is conjectured that m3(2, q), the size of the largest complete (k, 3)-arc in PG(2, q), exceeds the number of rational points on an elliptic curve. Here, a sufficient condition for its positive resolution is given using combinatorial analysis. Exploiting its structure as a (k, 3)-arc, the elliptic curve is then considered as a method of constructing cubic arcs and results governing completeness are established. Finally, classical theorems relating the order of the plane q to the existence of an elliptic curve with a specified number of rational points are used to extend theoretical results providing upper bounds to t3(2, q), the size of the smallest possible complete (k, 3)-arc in PG(2, q).

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In the projective space PG(k−1; q) over Fq, the finite field of order q, an (n; r)-arc K is a set of n points with at most r on a hyperplane and there is some hyperplane meeting K in exactly r points. An arc is complete if it is maximal with respect to inclusion. The arc K corresponds to a projective [n; k;n − r]q-code of length n, dimension k, and minimum distance n − r; if K is a complete arc, then the corresponding projective code cannot be extended. In this thesis, the n-sets in PG(1; 19) up to n = 10 and the n-arcs in PG(2; 19) for 4 B n B 20 in both the complete and incomplete cases are classified. The set of rational points of a non-singular, plane cubic curve can be considered as an arc of degree three. Over F19, these curves are classified, and the maximum size of the complete arc of degree three that can be constructed from each such incomplete arc is given.

A binary low-density parity-check (LDPC) code is a linear block code that is defined by a sparse parity-check matrix H, that is H has a low density of 1's. LDPC codes were originally presented by Gallager in his doctoral dissertation [9], but largely overlooked for the next 35 years. A notable exception was [29], in which Tanner introduced a graphical representation for LDPC codes, now known as Tanner graphs. However, interest in these codes has greatly increased since 1996 with the publication of [22] and other papers, since it has been realised that LDPC codes are capable of achieving near-optimal performance when decoded using iterative decoding algorithms. LDPC codes can be constructed randomly by using a computer algorithm to generate a suitable matrix H. However, it is also possible to construct LDPC codes explicitly using various incidence structures in discrete mathematics. For example, LDPC codes can be constructed based on the points and lines of finite geometries: there are many examples in the literature (see for example [18, 28]). These constructed codes can possess certain advantages over randomly-generated codes. For example they may provide more efficient encoding algorithms than randomly-generated codes. Furthermore it can be easier to understand and determine the properties of such codes because of the underlying structure. LDPC codes have been constructed based on incidence structures known as partial geometries [16]. The aim of this research is to provide examples of new codes constructed based on structures known as semipartial geometries (SPGs), which are generalisations of partial geometries. Since the commencement of this thesis [19] was published, which showed that codes could be constructed from semipartial geometries and provided some examples and basic results. By necessity this thesis contains a number of results from that paper. However, it should be noted that the scope of [19] is fairly limited and that the overlap between the current thesis and [19] is consequently small. [19] also contains a number of errors, some of which have been noted and corrected in this thesis.

In this thesis we present a range of different knot theories and then generalise them. Working with this, we focus on biquandles with linear and quadratic biquandle functions (in the quadratic case we restrict ourselves to functions with commutative coefficients). In particular, we show that if a biquandle is commutative, the biquandle function must have non-commutative coefficients, which ties in with the Alexander biquandle in the linear case. We then describe some computational work used to calculate rack and birack homology.

In PG(2; q), the projective plane over the field Fq of q elements, a (k; n)-arc is a set K of k points with at most n points on any line of the plane. When n = 2, a (k; 2)-arc is called a k-arc. A fundamental question is to determine the values of k for which K is complete, that is, not contained in a (k + 1; n)-arc. In particular, what is the largest value of k for a complete K, denoted by mn(2; q)? This thesis focusses on using some algorithms in Fortran and GAP to find large com- plete (k; n)-arcs in PG(2; q). A blocking set B is a set of points such that each line contains at least t points of B and some line contains exactly t points of B. Here, B is the complement of a (k; n)-arc K with t = q +1 - n. Non-existence of some (k; n)-arcs is proved for q = 19; 23; 43. Also, a new largest bound of complete (k; n)-arcs for prime q and n > (q-3)/2 is found. A new lower bound is proved for smallest size of complete (k; n)-arcs in PG(2; q). Five algorithms are explained and the classification of (k; n)- arcs is found for some values of n and q. High performance computing is an important part of this thesis, where Algorithm Five is used with OpenMP that reduces the time of implementation. Also, a (k; n)-arc K corresponds to a projective [k; n; d]q-code of length k, dimension n, and minimum distance d = k - n. Some applications of finite geometry to operational research are also explained.

In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19.

A major theme in geometric measure theory is establishing global properties, such as rectifiability, of sets or measures from local ones, such as densities or tangent measures. In establishing sufficient conditions for rectifiability it is useful to know what local properties are possible in a given setting, and this is the theme of this thesis. It is known, for 1-dimensional subsets of the plane with positive lower density, that the tangent measures being concentrated on a line is sufficient to imply rectifiability. It is shown here that this cannot be relaxed too much by demonstrating the existence of a 1-dimensional subset of the plane with positive lower density whose tangent measures are concentrated on the union of two halflines, and yet the set is unrectiable. A class of metrics are also defined on R, which are functions of the Euclidean metric, to give spaces of dimension s (s > 1), where the lower density is strictly greater than 21-s, and a method for gaining an explicit lower bound for a given dimension is developed. The results are related to the generalised Besicovitch 1/2 conjecture. Set functions are defined that measure how easily the subsets of a set can be covered by balls (of any radius) with centres in the subset. These set functions are studied and used to give lower bounds on the upper density of subsets of a normed space, in particular Euclidean spaces. Further attention is paid to subsets of R, where more explicit bounds are given.

The projective plane of order 11 is the dominant focus of this work. The motivation for working in the projective plane of order 11 is twofold. First, it is the smallest projective plane of prime power order such that the size of the largest (n, r)-arc is not known for all r ∈ {2,...,q + 1}. It is also the smallest projective plane of prime order such that the (n; 3)-arcs are not classified. Second, the number of (n, 3)-arcs is significantly higher in the projective plane of order 11 than it is in the projective plane of order 7, giving a large number of (n, 3)-arcs for study. The main application of (n, r)-arcs is to the study of linear codes. As a forerunner to the work in the projective plane of order eleven two algorithms are used to raise the lower bound on the size of the smallest complete n-arc in the projective plane of order thirty-one from 12 to 13. This work presents the classification up to projective equivalence of the complete (n, 3)- arcs in PG(2, 11) and the backtracking algorithm that is used in its construction. This algorithm is based on the algorithm used in [3]; it is adapted to work on (n, 3)-arcs as opposed to n-arcs. This algorithm yields one representative from every projectively inequivalent class of (n, 3)-arc. The equivalence classes of complete (n, 3)-arcs are then further classified according to their stabilizer group. The classification of all (n, 3)-arcs up to projective equivalence in PG(2, 11) is the foundation of an exhaustive search that takes one element from every equivalence class and determines if it can be extended to an (n′, 4)-arc. This search confirmed that in PG(2, 11) no (n, 3)-arc can be extended to a (33, 4)-arc and that subsequently m4(2, 11) = 32. This same algorithm is used to determine four projectively inequivalent complete (32, 4)-arcs, extended from complete (n, 3)-arcs. Various notions under the general title of symmetry are defined both for an (n, r)-arc and for sets of points and lines. The first of these makes the classification of incomplete (n; 3)- arcs in PG(2, 11) practical. The second establishes a symmetry based around the incidence structure of each of the four projectively inequivalent complete (32, 4)-arcs in PG(2, 11); this allows the discovery of their duals. Both notions of symmetry are used to analyze the incidence structure of n-arcs in PG(2, q), for q = 11, 13, 17, 19. The penultimate chapter demonstrates that it is possible to construct an (n, r)-arc with a stabilizer group that contains a subgroup of order p, where p is a prime, without reference to an (m < n, r)-arc, with stabilizer group isomorphic to ℤ1. This method is used to find q-arcs and (q + 1)-arcs in PG(2, q), for q = 23 and 29, supporting Conjecture 6.7. The work ends with an investigation into the effect of projectivities that are induced by a matrix of prime order p on the projective planes. This investigation looks at the points and subsets of points of order p that are closed under the right action of such matrices and their structure in the projective plane. An application of these structures is a restriction on the size of an (n, r)-arc in PG(2, q) that can be stabilized by a matrix of prime order p.

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