# to get a MDS code from a hyperoval in a projective plane

explain how we can get a MDS code of length q+2 and dimension q-1 from a hyperoval

in a projective plane PG2(q) with q a power of 2?

HINT:a hyperoval Q is a set of q+2 points such that no three points in Q are collinear.

you are expected to get a [q+2,q-1,4] binary code from this.take points,one dimensional

subspaces and blocks as the lines 2-dimensional subspaces

The standard way to get projective coordinates of the points of a hyperoval over $\mathbb F_q$ is to take the vectors $[1 : t : t^2]$ with $t\in\mathbb F_q$ together with $[0 : 1 : 0]$ and $[0 : 0 : 1]$.
Placing these vectors into the columns of a matrix, in the example $\mathbb F_4 = \{0,1,a,a^2\}$ (with $a^2 + a + 1 = 0$) a possible check matrix of a $[6,3,4]$ MDS code is $$\begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & a & a^2 \\ 0 & 0 & 1 & 1 & a^2 & a \end{pmatrix}$$ This specific code is also called Hexacode.
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.
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.