  In this article we prove that a class of Goppa codes whose Goppa polynomial
is of the form $g(x) = x + x^q + \cdots + x^{q^{m-1}}$ where $m \geq 3$ (i.e.
$g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a
better minimum distance than what the Goppa bound $d \geq 2deg(g(x))+1$
implies. Our improvement is based on finding another Goppa polynomial $h$ such
that $C(L,g) = C(M, h)$ but $deg(h) > deg(g)$. This is a significant
improvement over Trace Goppa codes over quadratic field extensions (i.e. the
case $m = 2$), as the Goppa bound for the quadratic case is sharp.
