  The higher rank numerical range is closely connected to the construction of
quantum error correction code for a noisy quantum channel. It is known that if
a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher
rank numerical range $\Lambda_k(A)$ is the intersection of convex polygons with
vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1}
\le n$. In this paper, it is shown that the higher rank numerical range of a
normal matrix with $m$ distinct eigenvalues can be written as the intersection
of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex
polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$
with minimum $n$ such that $\Lambda_k(A) = {\mathcal P}$. In particular, if
${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A
\in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $\Lambda_k(A) =
{\mathcal P}$.
