Codes from Veronese and Segre Embeddings and Hamadaʹs Formula

In this article we study the codes given by l hypersurfaces in Pnq to obtain a new formula for the dimension of codes given by (n−l) flats. We also obtain a new formula for the dimension of the ν th order generalized Reed–Muller code and describe the code given by the hyperplane intersections of the Segre embedding of Pnq×Pmq.