  A permutation-invariant quantum code on $N$ qudits is any subspace stabilized
by the matrix representation of the symmetric group $S_N$ as permutation
matrices that permute the underlying $N$ subsystems. When each subsystem is a
complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code
is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an
algebraic construction of new families of of $d$-dimensional
permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also
correct $t$ errors for $d \ge 2$. The construction of our codes relies on a
real polynomial with multiple roots at the roots of unity, and a sequence of
$q-1$ real polynomials that satisfy some combinatorial constraints. When $N >
(2t+1)^2(d-1)$, we prove constructively that an uncountable number of such
codes exist.
