Hint: Form a check matrix with $q+2$ columns and three rows. Place the homogeneous coordinates of the points on the hyperoval into the columns. Prove that the code with this check matrix has no words of weight $\le3$. The points of the projective plane are distinct => no words of weight two. No three collinear => no words of weight three.
More precisely. If there is a word of weight two or three (or any number, really), this means that those two or three (or whatever) columns of the check matrix must be linearly dependent. Here the presence of two linearly dependent columns means that the two points on the hyperoval are equal. The presence of three linearly dependent columns means that the homgeneous coordinates of the corresponding three points satisfy a linear equation, i.e. they are collinear.
Comment: You cannot expect to get a binary code (the Griesmer bound forbids the existence of a binary code with these parameters when $q>2$). You do get a $q$-ary code, which is presumably want the question really asks about.