Permutation-invariant qudit codes from polynomials

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 $({\mathbb{C}}^q{)}^N\mathrm{.}$ 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.