An investigation of an open case of the famous conjecture made by Hamada is carried out in the first part of this dissertation. In 1973, Hamada made the following conjecture: Let D be a geometric design having as blocks the d-subspaces of PG( n,q) or AG(n,q), and let m be the p-rank of D. If D' is a design with the same parameters as D, then the p-rank of D' is greater or equal to m, and equality holds if and only if D' is isomorphic to D. In 1986, Tonchev, and more recently Harada, Lam and Tonchev, Jungnickel and Tonchev, and Clark, Jungnickel and Tonchev found designs having the same parameters and p-rank as certain geometric designs, hence providing counter-examples to the "only-if" part of Hamada's conjecture. We discuss some properties of the three known nonisomorphic 2-(64,16,5) designs of 2-rank 16, one being the design of the planes in the 3-dimensional affine geometry over the field of order 4. We also try to find an algebraic way to use the similarities between these designs in a search for counter-examples to Hamada's conjecture in affine spaces of higher dimension.;Currently we know the existence of 22 projective planes of order 16 up to isomorphism, of which 4 are self dual. In the second part of this thesis, details of 2-(52,4,1) designs associated with known maximal 52-arcs are provided. A number of new maximal (52,4)-arcs in two of the known projective planes of order 16 are established. Newly discovered maximal (52,4)-arcs give new connections between the projective planes of order 16 as well. Previously the number of pairwise non-isomorphic resolutions of 2-(52,4,1) designs was ≥ 30. With the results in Tables 4.2 and 4.3, this bound is improved. Details of partial geometries coming from known maximal 52-arcs (including ours) are summarized in Table 4.4. It was pointed out that 104-sets of type (4,8) might arise from the unions of two disjoint maximal (52,4)-arcs. We detail the discovery of 37 new 104-sets of type (4,8), 18 of which come from unions of non-isomorphic maximal 52-arcs. Previous to our work, no such examples were known to exist. We discovered that the Johnson plane also contains disjoint maximal 52-arcs. Previously no such sets in the Johnson plane were known.
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