Partitions by subgeometries of projective plane over a galois field of order 3n, n = 1, 2, 3

Abstract

The purpose of this thesis is to split the projective plane of PG (2,9) into seven disjoint projective subplanes PG (2,3) and thirteen disjoint complete arcs of degree two and size seven. The projective line of order twenty-seven PG (1,27) has been partitioned into seven disjoint projective sublines PG (1,3) and the number of inequivalent -sets which are unordered sets of four points is classified. The group action on projective lines PG (1,3ⁿ) and projective PG (1,3ⁿ), n = 1, 2, 3 planes is explained and we have introduced theorems and examples and the subspaces of PG (1,3ⁿ)  and PG (2,3ⁿ), n = 1, 2, 3  are shown. Each of these partitions gives rise to an error-correcting code that corrects the maximum possible number of errors for its length. GAP-Groups–Algorithms, Programming a system for computational discrete algebra has been applied.