Classification of small triorthogonal codes

Triorthogonal codes are a class of quantum error correcting used in magic state distillation protocols. We classify all triorthogonal with $n+k \le 38$, where $n$ is the number physical qubits and $k$ logical code. find $38$ distinguished subspaces show that every code $n+k\le 38$ descends from one these through elementary operations such as puncturing deleting qubits. Specifically, we associate each Reed-Muller polynomial weight $n+k$, polynomials low using results Kasami, Tokura, Azumi an extensive computerized search. In appendix independent main text, improve protocol by reducing time variance due to stochastic Clifford corrections.