Some Applications of Magma in Designs and Codes: Oval Designs, HermitianUnitals and Generalized Reed–Muller Codes

We describe three applications of Magma to problems in the area of designs and the associated codes: • Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs; • the Hermitianunital as a 2-(q3 + 1, q + 1, 1) design, and the code overFp where p divides q + 1; • a basis of minimum-weight vectors for the code over Fpof the design of points and hyperplanes of the affine geometry AGd(Fp), where p is a prime.