  In 1999, Iwan Duursma defined the zeta function for a linear code as a
generating function of its Hamming weight enumerator. It can also be defined
for other homogeneous polynomials not corresponding to existing codes. If the
homogeneous polynomial is invariant under the MacWilliams transform, then its
zeta function satisfies a functional equation and we can formulate an analogue
of the Riemann hypothesis. As far as existing codes are concerned, the Riemann
hypothesis is believed to be closely related to the extremal property.
  In this article, we show there are abundant polynomials invariant by the
MacWilliams transform which satisfy the Riemann hypothesis. The proof is
carried out by explicit construction of such polynomials. To prove the Riemann
hypothesis for a certain class of invariant polynomials, we establish an
analogue of the Enestr"om-Kakeya theorem.
