Higher rank numerical ranges of normal matrices

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 ε Mn has eigenvalues a₁, ⋯ , a _n, then its higher rank numerical range Γ_κ(A) is the intersection of convex polygons with vertices a_j1 , ⋯ , a_jn-k+1, where 1 ≤ j₁ < ⋯ ≤ j _n-k+1 ≤ 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 P, a construction is given for a normal matrix A ε M_n with minimum n such that Δ_κ(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ε M_n with n ≤ max {p + k -1, 2k + 2} such that Γ_κ(A) = P.