The number of solutions of the equation image and some applications

We compute the number of solutions of the equation image in image, where f denote a quadratic hermitian form on image, image and image, and we deduce the number of hermitian matrices of order N and rank ρ. This number is well-known since the paper of Carlitz and Hodges (Duke Math. J. 22 (1995) 393), but with a more restrictive definition of hermitian matrices and with a rather different proof. Next, we introduce a linear code Γ(N,t,s) on image constructed with the same method as Reed–Muller one, and compute its weight distribution. Γ(N,t,s) is a generalization of the two codes Γ and C studied in Mercier (J. Pure Appl. Algebra 173 (3) (2003) 273) and the method for obtaining its weight distribution is new and more straightforward. Tools are exponential sums and linear algebra on image.