发表机构
DGIST（大邱科学技术院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明秩度量CSS码的非对称Singleton界并刻画等式结构，揭示堆叠架构中稳定子码的精确投影恢复半径优势，并构造达到最优解码半径的显式码族。
AI 中文摘要
来自共享控制的关联故障可能具有密集的物理支撑，但在基域上秩较低，这推动了堆叠架构中秩度量量子码的研究。我们证明了Calderbank--Shor--Steane码（包括退化码）的非对称秩度量Singleton界，并通过交换的最大秩距离矩阵码刻画了等式成立的条件。对于每个允许的参数三元组，最优对都存在且必然是纯的。与擦除界的比较表明，在特定的高布局下，固定物理资源时，一般稳定子码在堆叠秩距离上具有精确优势。我们确定了放宽恢复目标是否会扩大最坏情况可纠正半径。对于具有正逻辑维数的Singleton最优对，在严格低于扇区秩距离一半的半径上，在每个布局和每个非零投影子上，测量的综合征精确地确定投影误差模稳定子。当互补校验空间在伴随投影子下不变时，投影综合征在任意半径上可恢复；否则，适用相同的半径限制。对于非平凡幂等元，这种不变性要求行数多于列数或扇区距离为一。对于迹自伴投影子，当且仅当码在投影子上分裂为张量积时，两个扇区中存在线性环境投影综合征接口。将两个校验空间置于互补投影子图像中的设计在编码逻辑信息时必然是一侧的。两个显式族实现了等式结构的极端情况，包括一个具有高效认证解码器并达到最优唯一解码半径的双侧族。
英文摘要
Correlated faults from shared control can have dense physical support yet low rank over a base field, motivating rank-metric quantum codes for stacked architectures. We prove an asymmetric rank-metric Singleton bound for Calderbank--Shor--Steane codes, including degenerate codes, and characterize equality through commuting maximum-rank-distance matrix codes. Optimal pairs exist for every admissible parameter triple and are necessarily pure. Comparison with erasure bounds establishes an exact advantage in stacked rank distance for general stabilizer codes at fixed physical resources on certain tall layouts. We determine whether relaxing the recovery target enlarges the worst-case correctable radius. For Singleton-optimal pairs with positive logical dimension, the measured syndrome determines the projected error modulo stabilizers exactly at radii strictly below half the sector rank distance, on every layout and for every nonzero projector. The projected syndrome is recoverable at arbitrary radii exactly when the complementary check space is invariant under the adjoint projector; otherwise, the same radius limit applies. For nontrivial idempotents, this invariance requires more rows than columns or sector distance one. For trace-self-adjoint projectors, linear ambient projected-syndrome interfaces in both sectors exist exactly when the code splits as a tensor product across the projector. Designs placing the two check spaces in complementary projector images are necessarily one-sided whenever they encode logical information. Two explicit families realize the extreme cases of the equality structure, including a two-sided family with an efficient certifying decoder attaining the optimal unique-decoding radius.
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