A family of permutationally invariant quantum codes

We construct a new family of permutationally invariant codes that correct $t$ Pauli errors for any $t\ge 1$. We also show that codes in the new family correct spontaneous decay errors as well as deletion errors. In many cases the codes in this family are shorter than the best previously known explicit families of permutationally invariant codes both for Pauli errors, deletions, and for the amplitude damping channel. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$ Pauli errors from the previously known results for $t=1$ to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer.