Here is a list of code families which contain pefect quantum codes.
Code Description
Five-qubit perfect code Five-qubit stabilizer code with generators that are symmetric under cyclic permutation of qubits, \begin{align} \begin{split} S_1 &= IXZZX \\ S_2 &= XZZXI \\ S_3 &= ZZXIX \\ S_4 &= ZXIXZ~. \end{split} \end{align}
Perfect code An $$(n,K,2t+1)_q$$ binary or $$q$$-ary code is perfect if parameters $$n$$, $$K$$, $$t$$, and $$q$$ are such that the Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t}(q-1)^{j}{n \choose j}\leq q^{n}/K \end{align} becomes an equality. For example, for a binary $$q=2$$ code with one logical bit ($$K=2$$) and $$t=1$$, the bound becomes $$n+1 \leq 2^{n-1}$$. Perfect codes are those for which balls of Hamming radius $$t$$ exactly fill the space of all $$n$$ $$q$$-ary strings.
Perfect quantum code A non-degenerate code constructed out of $$q$$-dimensional qudits and having parameters $$((n,K,2t+1))$$ is perfect if $$n$$, $$K$$, $$t$$, and $$q$$ are such that the quantum Hamming bound \begin{align} \sum_{j=0}^{t}(q^2-1)^{j}{n \choose j}\leq q^{n}/K \end{align} becomes an equality. For example, for a qubit $$q=2$$ code with one logical qubit ($$K=2$$) and $$t=1$$, the bound becomes $$3n+1 \leq 2^{n-1}$$. The bound can be saturated only at certain $$n$$.
$$[[15, 7, 3]]$$ Hamming-based CSS code Self-dual Hamming-based CSS code that admits permutation-based CZ logical gates.
$$[[2^r, 2^r-r-2, 3]]$$ quantum Hamming code A family of stabilizer codes of distance $$3$$ that saturate the asymptotic quantum Hamming bound. Can be obtained from the CSS construction using a first-order $$[2^r,r+1,2^{r-1}]$$ RM code and a $$[2^r,2^r-1,2]$$ even-weight code .

## References


A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996). DOI; quant-ph/9605021