Second highest number of points of hypersurfaces in

For generalized Reed–Muller, GRM(q,d,n), codes, the determination of the second weight is still generally unsolved, it is an open question of Cherdieu and Rolland [J.P. Cherdieu, R. Rolland, On the number of points of some hypersurfaces in , Finite Fields Appl. 2 (1996) 214–224]. In order to answer this question, we study some maximal hypersurfaces and we compute the second weight of GRM(q,d,n) codes with the restriction that q 2d.