Here is a list of all quantum codes useful for distilling magic states and characterized by their magic-state distillation scaling exponent.
Name Magic-state scaling exponent
H code A total of \(r\) rounds of magic-state distillation yields a magic-state scaling exponent \(\gamma\to 1\) as \(k,r\rightarrow \infty\). This matches a conjectured bound for \(\gamma\) [1].
Triorthogonal code Depends on the matrix. Reference [1] gave a family of \(\frac{k}{3k+8}\) codes with magic-state distillation scaling exponent \(\gamma = \log_2 \frac{3k+8}{k}\).
\([[15,1,3]]\) Reed-Muller code Magic-state distillation scaling exponent \( \gamma= \log_d (n/k)\approx 2.46\) [2].


S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017). DOI; 1703.07847