COMBINATORIAL PROBLEMS RELATED TO CODES, DESIGNS AND FINITE GEOMETRIES
Mustafa Gezek · 2017
An investigation of an open case of the famous conjecture made by Hamada \cite{Hamada1} 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 \textit{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 $D$. 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 the 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. Previously the number of pairwise non-isomorphic resolutions of 2-(52,4,1) designs was > 30. With the results in Tables 4.2-4.3, this bound is improved. Details of partial geometries coming from known maximal 52-arcs (including ours) are summarized in Table 4.4. 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.