Codes Defined by Forms of Degree $2$ on Quadric Surfaces

In this correspondence, we study the functional codes C
₂
(X) defined on projective varieties X, in the case where X sub P
³
(F
_q
) is a 1-degenerate quadric or a nondegenerate quadric (hyperbolic or elliptic). We find the minimum distance of these codes, the second weight, and the third weight. We also show the geometrical structure of the first weight and second weight codewords. One result states that the codes C
₂
(X) defined on the elliptic quadrics are good codes according to the table of A. E. Brouwer.