A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground-space is permutation-invariant. Hence we go beyond the stabilizer formalism for quantum codes and consider permutation-invariant quantum codes. Using techniques from combinatorics and operator theory, for any positive integer t, we construct families of permutation-invariant quantum codes of length proportional to t2 that that perfectly correct arbitrary weight t errors and other families of permutation-invariant quantum codes approximately correct t spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.