Evaluation codes from smooth quadric surfaces and twisted Segre varieties

We give the parameters of any evaluation code on a smooth quadric surface. For hyperbolic quadrics the approach uses elementary results on product codes and the parameters of codes on elliptic quadrics are obtained by detecting a BCH structure on these codes and using the BCH bound. The elliptic quadric is a twist of the surface P ¹ × P ¹ and we detect a similar BCH structure on twists of the Segre embedding of a product of any d copies of the projective line.