Permutation-invariant qudit codes from polynomials

Abstract 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 ≥ 2 , any permutation-invariant code is a subspace of the symmetric subspace of ( C q ) N . We give an algebraic construction of new families of d -dimensional permutation-invariant codes on at least ( 2 t + 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 > ( 2 t + 1 ) 2 ( d − 1 ) , we prove constructively that an uncountable number of such codes exist.