Perfect quantum error-correcting code ($2^k{\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)=2^n$)

A nondegenerate quantum code is perfect at correction radius $t$ if its distinct correctable Pauli error images exhaust the physical Hilbert space. Each image has dimension $2^k$, and their orthogonal direct sum is the whole $2^n$-dimensional space precisely when the quantum Hamming bound is an equality. This perfectness concerns unknown-location errors of weight at most $t$; it does not imply correction of every higher-weight error.