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 SN as permutation matrices that permute the underlying N subsystems. When each subsystem is a complex Euclidean space of dimension q≥2, any permutation-invariant code is a subspace of the symmetric subspace of (Cq)N. We give an algebraic construction of new families of d-dimensional permutation-invariant codes on at least (2t+1)2(d−1) qudits that can also correct t errors for d≥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.