A triorthogonal code is a binary quantum Calderbank-Shor-Steane (CSS) code defined by a triorthogonal matrix. Triorthogonal codes are a key ingredient in magic-state distillation, since they allow for transversal $\mathsf{T}$ gates, a non-Clifford logical operation useful for achieving universal fault-tolerant quantum computation. Their construction is challenging because it must satisfy simultaneous pairwise and triple-wise overlap constraints, as well as row-weight requirements. In this work, we study the construction and decoding of triorthogonal codes with prescribed 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 new nontrivial triorthogonal codes that are not necessarily generated by classical triply-even codes. The decoding performance of high-distance triorthogonal codes obtained via the doubling construction is then evaluated over the dephasing channel. We compare bounded-distance decoding, belief propagation plus ordered-statistics post-processing, and a GRAND-based decoder adapted to the quantum setting, which turns out to be a promising option.

翻译：三正交码是一类由三正交矩阵定义的二元量子Calderbank-Shor-Steane（CSS）码。三正交码是魔法态蒸馏的关键组成部分，因为它们允许实现横向$\mathsf{T}$门——一种有助于实现通用容错量子计算的非克利福德逻辑操作。其构造具有挑战性，因为必须同时满足成对及三重重叠约束以及行权重需求。本文研究具有指定对偶距离性质的三正交码的构造与解码问题。我们推导出具有目标对偶最小距离的偶权重三正交生成矩阵的存在性判据。该判据通过Krawtchouk多项式条件将三正交约束与MacWilliams恒等式相结合，作用于对偶权重分布，从而将构造问题转化为整数线性规划形式。我们发现了新的非平凡三正交码，这些码不一定由经典的三偶码生成。随后，通过加倍构造得到的高距离三正交码在去相位信道上的解码性能得到评估。我们比较了有界距离解码、置信传播结合有序统计后处理以及一种适用于量子场景的基于GRAND的解码器，后者被证明是一种有前景的方案。