Lines on K3 quartic surfaces in characteristic 3

We investigate the number of straight lines contained in a K3 quartic surface $X$ defined over an algebraically closed field of characteristic 3. We prove that if $X$ contains 112 lines, then $X$ is projectively equivalent to the Fermat quartic surface; otherwise, $X$ contains at most 67 lines. We improve this bound to 58 if $X$ contains a star (ie four distinct lines intersecting at a smooth point of $X$). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.