Group Structure on Projective Spaces and Cyclic Codes over Finite Fields

We study the geometrical properties of the subgroups of the mutliplicative group of a finite extension of a finite field endowed with its vector space structure and we show that in some cases the associated projective space has a natural group structure. We construct some cyclic codes related to Reed–Muller codes by evaluating polynomials on these subgroups. The geometrical properties of these groups give a fairly simple description of these codes which are of the Reed–Muller kind.