arXiv:2605.24519v2 Announce Type: replace-cross Abstract: Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical $\mat{T}$ gates. As a non-Clifford operation, the $\mat{T}$ gate enables universal fault-tolerant quantum computation when combined with Clifford operations. In this work, we study the construction and decoding of triorthogonal codes with given dual distance properties. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate), that are not necessarily constructed from triply-even binary linear codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding and guessing random additive-noise decoding (GRAND) are promising options.