发表机构
Department of Physics and Computer Science, Dayalbagh Educational Institute（Dayalbagh 教育学院物理与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究素数维数下的循环三正交码，利用谱支撑和集条件进行穷举搜索，并证明三偶性与经典三正交性等价，找到优于现有码的短码及无限码族。
AI 中文摘要
我们研究了素数维数 $p$ 下的循环三正交码，在这些码上，一个横向的第三级对角门充当逻辑非克利福德门。对于奇数 $p$，我们使用一个不需要自正交性的三正交性定义。对于长度与 $p$ 互素的循环码，三正交性归结为谱支撑上的一个和集条件，这使得穷举搜索可行：对于两种自然构造，我们枚举了 $p=3$ 时长度至 $121$、$p=5,7$ 时长度至 $96$ 的所有此类极大码。对于量子比特，我们证明奇数长度的循环码是三偶的当且仅当它在经典意义上是三正交的，从而给出了三偶循环码的完整描述，并确定了长度至 $763$ 的所有此类码。对于量子比特系统（qudits），我们找到了如 $[[19,1,3]]_3$ 和 $[[7,1,3]]_7$ 的码，这些码在开销指数和蒸馏阈值两方面都优于最短的里德-穆勒码和里德-所罗门码。后者是 $[[p,1,2\lfloor p/6\rfloor+1]]_p$ 码的无限族中的第一个成员。
英文摘要
We study cyclic triorthogonal codes in prime dimension $p$, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd $p$, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to $p$, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for $p=3$ up to length $121$ and $p=5,7$ up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as $[[19,1,3]]_3$ and $[[7,1,3]]_7$, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of $[[p,1,2\lfloor p/6\rfloor+1]]_p$ codes.
Comments24 pages, 2 tables. Code and data: https://github.com/spdei/cyclic-triorthogonal-codes
