On the numerical range of matrices defined over a finite field

Let q be a prime power. For u = (u(1), ..., u(n)),v = (v(1), ..., v(n)) is an element of F-q2(n) let < u, v > : = Sigma(n)(i=1) u(i)(q)v(i) be the Hermitian form of F-q2(n). Fix an n x n matrix M over F-q2. Set Num(M) : = {< u, Mu > vertical bar u is an element of F-q2(n), < u, u > = 1} (the numerical range of M introduced by Coons, Jenkins, Knowles, Luke and Rault (case q a prime q equivalent to 3 (mod 4)) and by the author (arbitrary q)). When n = 2 we prove an upper bound for vertical bar Num(M)vertical bar. We describe Num(M) for several classes of matrices, mostly for n = 2, 4. (C) 2020 Elsevier Inc. All rights reserved.