Here is a list of all quantum codes that specify what rate they have.
Name Rate
Alamouti code The only OSTBC with unity rate.
Bacon-Shor code A non-LDPC family of Bacon-Shor codes achieves a distance of $$\Omega(n^{1-\epsilon})$$ with sparse gauge operators.
Balanced product code A notable family of balanced product codes encode $$k \in \Theta(n^{4/5})$$ logical qubits with distance $$d \in \Omega(n^{3/5})$$ for any number of physical qubits $$n$$. Additionally, it is known that the code constructed from the balanced product of two good classical LDPC codes over groups of order $$\Theta(n)$$ has a constant encoding rate [1].
Binary Golay code The perfect binary Golay code has a rate of $$12/23 = 0.522$$. The extended binary Golay code has a rate of $$12/24 = 0.5$$.
Binary repetition code Code rate is $$\frac{1}{n}$$, code distance is $$n$$.
Chuang-Leung-Yamamoto code Code rate is $$\frac{k}{n \log_2(N+1)}$$. To correct the loss of up to $$t$$ excitations with $$K+1$$ codewords, a code exists with scaling $$N \sim t^3 K/2$$.
Color code For general 2D manifolds, $$kd^2 \leq c(\log k)^2 n$$ for some constant $$c$$ [2], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in $$n$$.
Constant-excitation (CE) code Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the pure-loss Gaussian channel [3].
Convolutional code Depends on the polynomials used. Using puncturing removal [4] the rate for the code can be increased from $$\frac{1}{t}$$ to $$\frac{s}{t}$$, where $$t$$ is the number of output bits, and $$s$$ depends on the puncturing done. This is done by deleting some pieces of the encoder output such that the most-likely decoders remain effective
Expander code The rate is $$1 - m/n$$ where $$n$$ is the number of left nodes and $$m$$ is the number of right nodes in the bipartite expander graph.
Expander lifted-product code Expander lifted-product codes include the first examples [5] of (asymptotically) good QLDPC codes, i.e., codes with asymptotically constant rate and linear distance. The existence of such codes proves the QLDPC conjecture [6]. Another notable family encodes $$k \in \Theta(n^\alpha \log n)$$ logical qubits with distance $$d \in \Omega(n^{1 - \alpha} / \log n)$$ for any number of physical qubits $$n$$ and any real parameter $$0 \leq \alpha < 1$$ [7].
Fiber-bundle code Rate $$k/n = \Omega(n^{-2/5}/\text{polylog}(n))$$, distance $$d=\Omega(n^{3/5}/\text{polylog}(n))$$. This is the first QLDPC code to achieve a distance scaling better than $$\sqrt{n}~\text{polylog}(n)$$.
Fountain code Random linear fountain codes approach the Shannon limit as the file size $$K$$ increases.
Freedman-Meyer-Luo code Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, $$d=\Omega(\sqrt{n \sqrt{\log n}})$$, broken by Ramanujan tensor-product codes.
Goppa code There exist Goppa codes defined over larger alphabets that meet the Gilbert-Varshamov, or GV, bound.
H code The H codes are dense, i.e., the rate $$\frac{k}{k+4}\rightarrow 1$$ as $$k \rightarrow \infty$$. The distance is 2. However an $$r$$-level concatenation of H codes gives a distance of $$2^r$$.
Haar-random code The rate of the code is equal to the coherent information of the channel (i.e. the quantum channel capacity).
Hamming code Asymptotic rate $$k/n = 1-\frac{\log n}{n} \to 1$$ and normalized distance $$d/n \to 0$$.
Heavy-hexagon code $$1/n$$ for a distance-$$d$$ heavy-hexagon code on $$n = (5d^2-2d-1)/2$$ qubits.
Justesen code

The first asymptotically good codes. Rate is $$R_m=k/n=K/2N\geq R$$ and the relative minumum distance satisfy $$\delta_m=d_m/n\geq 0.11(1-2R)$$, where $$K=\left\lceil 2NR\right\rceil$$ for asymptotic rate $$0<R<1/2$$; see pg. 311 of Ref. [8].

The code can be improved and extend the range of $$R$$ from 0 to 1 by puncturing, i.e., by erasing $$s$$ digits from each inner codeword. This yields a code of length $$n=(2m-s)N$$ and rate $$R=mk/(2m-s)N$$. The lower bound of the distance of the punctured code approaches $$d_m/n=(1-R/r)H^{-1}(1-r)$$ as $$m$$ goes to infinity, where $$r$$ is the maximum of 1/2 and the solution to $$R=r^2/(1+\log(1-h^{-1}(1-r)))$$, and $$h$$ is the binary entropy function.

Kitaev surface code Rate depends on the underlying cellulation and manifold. For general 2D manifolds, $$kd^2\leq c(\log k)^2 n$$ for some constant $$c$$ [2], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as $$O(\sqrt{n})$$ [9][10], and (2) surface codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in $$n$$. Higher-dimensional hyperbolic manifolds (see code children below) yield distances scaling more favorably. Loewner's theorem provides an upper bound for any bounded-geometry surface code [11].
Lifted-product (LP) code There is no known simple way to compute the logical dimension $$k$$ in the general case [7].
Linear binary code A family of linear codes $$C_i = [n_i,k_i,d_i]$$ is asymptotically good if the asymptotic rate $$\lim_{i\to\infty} k_i/n_i$$ and asymptotic distance $$\lim_{i\to\infty} d_i/n_i$$ are both positive.
Locally recoverable code (LRC) The rate $$r$$ of an $$(n,k,r)$$ LRC code satisfies \begin{align} \frac{k}{n}\leq\frac{r}{r+1}~. \end{align}
Monitored random-circuit code Rate can be finite [12], depending on the family of random codes generated by the circuit.
Movassagh-Ouyang Hamiltonian code The rate depends on the classical code, but distance can scale linearly with $$n$$.
Multi-mode GKP code Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of displacement-noise and thermal-noise Gaussian loss channels [13][14][15][16].
Orthogonal Spacetime Block Code (OSTBC) The greatest rate which can be achieved is $$\frac{n_0+1}{2n_0}$$, where either $$n=2n_0$$ or $$n=2n_0-1$$ [17].
Pastawski-Yoshida-Harlow-Preskill (HaPPY) code The pentagon HaPPY code has an asymptotic rate $$\frac{1}{\sqrt{5}} \approx 0.447$$. The pentagon/hexagon HaPPY code, with alternating layers of pentagons and hexagons in the tiling, has a rate of $$0.299$$ if the last layer is a pentagon layer and a rate of $$0.088$$ if the last layer is a hexagon layer.
Perfect quantum code $$k/n\to 1$$ asymptotically with $$n$$.
Polar code Supports reliable transmission at rates $$K/N$$ approaching the Shannon capacity of the channel.
Projective-plane surface code The rate is $$1/n$$, where $$n$$ is the number of edges of the particular cellulation.
Quantum Reed-Muller code $$\frac{k}{n}$$, where $$k = 2^r - {r \choose t} + 2 \sum_{i=0}^{t-1} {r \choose i}$$. Additionally, CSS codes formed from binary Reed-Muller codes achieve channel capacity on erasure channels [18].
Quantum Tanner code Good QLDPC codes.
Quantum expander code $$[[n,k=\Theta(n),d=O(\sqrt{n})]]$$ code with asymptotically constant rate.
Quantum low-density parity-check (QLDPC) code A family of QLDPC codes $$[[n_i,k_i,d_i]]$$ is asymptotically good if the asymptotic rate $$\lim_{i\to\infty} k_i/n_i$$ and asymptotic distance $$\lim_{i\to\infty} d_i/n_i$$ are both positive. The first examples of good qubit codes are a family of lifted-product codes.
Ramanujan-complex product code For 2D Ramanujan complexes, the rate is $$\Omega(\sqrt{ \frac{1}{n \log n} })$$, with minimum distance $$d = \Omega(\sqrt{n \log n})$$. For 3D, the rate is $$\Omega(\frac{1}{\sqrt{n}\log n})$$ with minimum distance $$d \geq \sqrt{n} \log n$$.
Random code Typical random codes (TRC) or typical random linear codes (TLC) refer to codes in the respective ensemble that satisfy a certain minimum distance. The relative fraction of typical codes in the ensemble approaches one as $$N$$ goes to infinity [19] (see also Ref. [20]). Asymptotically, given distance $$d$$, the maximum rate for a TRC is given by $$R=\frac{1}{2}R_{GV}(\delta)$$ where $$R_{GV}$$ is the Gilbert–Varshamov (GV) bound $$R_{GV}=1-h(\delta)$$, and $$h(\delta)=h(\frac{d}{n})$$ is the binary entropy function. The maximum rate for a TLC is given by $$R=R_{GV}(d)$$, meaning that TLCs achieve the asymoptic GV bound.
Rank-modulation code Rank modulation codes with code distance $$d=\Theta(n^{1+\epsilon})$$ for $$\epsilon\in[0,1]$$ achieve a rate of $$1-\epsilon$$ [21].
Reed-Muller (RM) code Achieves capacity of the binary erasure channel [22].
Single parity-check code The code rate is $$\frac{n}{n+1}\to 1$$ as $$n\to\infty$$. The code distance is 2.
Tanner code For a short code $$C_0$$ with rate $$R_0$$, the Tanner code has rate $$R \geq 2R_0-1$$. If $$C_0$$ satisfies the Gilbert-Varshamov bound, the rate $$R \geq \delta = 1-2h(\delta_0)$$, where $$\delta$$ ($$\delta_0$$) is the relative distance of the Tanner code ($$C_0$$), and $$h$$ is the binary entropy function.
Tornado code Come arbitrarily close to the capacity of the binary erasure channel.
Two-dimensional hyperbolic surface code Two-dimensional hyperbolic surface codes have an asymptotically constant encoding rate $$k/n$$ with a distance scaling logarithmically with $$n$$ when the surface is closed. The encoding rate depends on the tiling $${r,s}$$ and is given by $$k/n = (1-2/r - 2/s) + 2/n$$, which approaches a constant value as the number of physical qubits grows. The weight of the stabilizers is $$r$$ for $$Z$$-checks and $$s$$ for $$X$$-checks. For open boundary conditions, the code reduces to constant distnace.
XYZ product code Not much has been proven about the relationship between XYZ-product codes and other codes. The logical dimension depends on properties of the input classical codes, specifically similarity invariants from abstract algebra. It is conjectured that specific instances of XYZ-product codes have a constant encoding rate and a minimum distance of $$d \in \Theta(n^{2/3})$$ [23].
Zetterberg code The rate is given by $$1-\frac{4s}{n}$$, which is asymptotically good, with a minimum distance of 5.

## References

[1]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021). DOI; 2012.09271
[2]
N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013). DOI; 1301.6588
[3]
Matthew S. Winnel et al., “Achieving the ultimate end-to-end rates of lossy quantum communication networks”. 2203.13924
[4]
L. Sari, “Effects of Puncturing Patterns on Punctured Convolutional Codes”, TELKOMNIKA (Telecommunication, Computing, Electronics and Control) 10, (2012). DOI
[5]
Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
[6]
N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309
[7]
P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022). DOI; 2012.04068
[8]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977
[9]
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010). DOI; 0909.5200
[10]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
[11]
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002). DOI
[12]
M. J. Gullans and D. A. Huse, “Dynamical Purification Phase Transition Induced by Quantum Measurements”, Physical Review X 10, (2020). DOI; 1905.05195
[13]
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001). DOI; quant-ph/0105058
[14]
K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018). DOI; 1708.07257
[15]
M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018). DOI; 1801.04731
[16]
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019). DOI; 1801.07271
[17]
Xue-Bin Liang, “Orthogonal designs with maximal rates”, IEEE Transactions on Information Theory 49, 2468 (2003). DOI
[18]
Shrinivas Kudekar et al., “Reed-Muller Codes Achieve Capacity on Erasure Channels”. 1601.04689
[19]
C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948). DOI
[20]
A. Barg and G. D. Forney, “Random codes: minimum distances and error exponents”, IEEE Transactions on Information Theory 48, 2568 (2002). DOI
[21]
A. Barg and A. Mazumdar, “Codes in permutations and error correction for rank modulation”, 2010 IEEE International Symposium on Information Theory (2010). DOI
[22]
S. Kudekar et al., “Reed–Muller Codes Achieve Capacity on Erasure Channels”, IEEE Transactions on Information Theory 63, 4298 (2017). DOI
[23]
Anthony Leverrier, Simon Apers, and Christophe Vuillot, “Quantum XYZ Product Codes”. 2011.09746