Recursive Construction of Triorthogonal Codes.

Triorthogonal codes are a class of quantum error-correcting codes that possess a transversal non-Clifford gates, and play a crucial role in an approach to fault-tolerant quantum computation magic state distillation. For near-term applications, it is of interest to identify as many small triorthogonal codes (of sizes less than approximately 100 qubits or qudits) as possible. To facilitate construction of such codes, we introduce the concept of a maximal triorthogonal space. We provide a a polynomial-time algorithm for checking whether a binary triorthgonal space is maximal. We find that triorthogonal spaces formed from binary Reed Muller codes of the form RM (r, 3r + j) are maximal if j = 1, but not maximal if j = 2 or 3. We also construct an efficient algorithm to

generate maximal triorthogonal spaces from non-maximal triorthogonal spaces, (although these spaces are not guaranteed to be optimal). Using this algorithm, we construct new triorthogonal codes with distance d ≥ 5 and size n ≤ 100.