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Constructing Hadamard Matrices from Jacobi Sums Bernhard Schmidt Nanyang Technological University, Singapore It is known that there are Hadamard matrices of order 4p4 for all primes p. The problem of constructing Hadamard matrices of order 4p for all primes p seems infeasible at present time, so we are looking at order 4p2. We survey the current knowledge in this case, and show how to use cyclotomy and Jacobi sums to settle the problem for two new families of primes p. The construction works for primes $p\equiv 7 \bmod 16$ satisfying certain diophantine equations. This is joint work with Ka Hin Leung and Siu Lun Ma.

Embeddings of Finite Partial Linear Spaces into Finite Projective Planes Jason Williford C U Denver In 1979 Paul Erdos posed the question of whether every finite partial linear space is embedded in a finite projective plane. In this talk we present a proof that every finite partial linear space is embedded in a finite translation net generated by a partial spread of an even dimensional vector space over an arbitrary finite field. Related results will also be discussed.