由经典 Reed–Muller 码构造的量子码族，其中 15-qubit 实例因非 Clifford transversal T 结构而常用于容错逻辑与 magic-state 方案。
记录码族本身及其 transversal/morphing 性质；具体 magic-state distillation 协议属于 logical 领域。
code/block/quantum-reed-mullerCode dossier
这里汇总参数景观中的结构化记录、码族谱系中的构造说明、执行路线与外部知识库，作为完整主题页的代码专属延伸。
n、k、d 与最大检查权重已收录；qubit degree 依赖稳定子生成元基，未选定统一基前不绘制六指标点。
15-qubit punctured quantum Reed–Muller CSS code。
15-qubit punctured quantum Reed–Muller CSS code。
classical RM construction
15-qubit instance
fault-tolerant T gate
记录码族本身及其 transversal/morphing 性质；具体 magic-state distillation 协议属于 logical 领域。
15-qubit Quantum Reed–Muller Execution Route：穿孔 Reed–Muller CSS 构造、分扇区纠错与横向非 Clifford 接口。当前路线绑定 Reed–Muller CSS decoder、Lookup decoder，并覆盖 CSS stabilizer extraction、Verified ancilla extraction。
n、k、d 与最大检查权重已收录；qubit degree 依赖稳定子生成元基，未选定统一基前不绘制六指标点。
横向 T 属于特定 punctured QRM 的多正交结构，不能推广到整个家族；码切换成本也必须计入逻辑门比较。
1999Quantum Reed-Muller codes2022Morphing Quantum CodesDeep reference
定义、代数构造、保护能力与执行证据均按来源段落独立维护；中文稿经过术语整理，英文稿保留用于逐段对照。
量子 Reed–Muller 码由经典 Reed–Muller 码及其对偶/嵌套关系构成 CSS 码族。不同文献的 puncturing、shortening 和阶数约定会改变参数；最常见的 15 量子比特码是具有横向 T 门的 [[15,1,3]] 实例。
Quantum Reed-Muller codes are CSS families built from nested classical Reed-Muller codes and their duals. Puncturing, shortening, and order conventions change parameters; the best-known instance is the [[15,1,3]] code with transversal T.
经典 RM(r,m) 由 m 变量、次数不超过 r 的布尔多项式取值向量生成。选择满足 CSS 包含关系的 X/Z 经典码并可能 puncture 一个坐标即可得到量子码；实例必须保存 r、m、puncture 集与生成元约定。
Classical RM(r,m) is generated by evaluation vectors of Boolean polynomials of degree at most r in m variables. Choosing nested X/Z codes and optionally puncturing coordinates yields a quantum CSS code; r, m, puncture set, and generator convention identify an instance.
距离由相关经典码及其对偶陪集中的最小重量决定，CSS 结构分别保护 X 与 Z 分量。[[15,1,3]] 只能纠正一个任意量子比特错误；横向非 Clifford 门并未提高其距离。
Distance is determined by minimum weights in the relevant classical codes and dual cosets, with CSS protection for X and Z components. The [[15,1,3]] code corrects one arbitrary qubit error; its transversal non-Clifford gate does not increase distance.
经典 RM 参数可解析计算，但量子参数还依赖嵌套、puncturing 和是否使用子系统扩展。报告应给出完整构造元组；只写“QRM(m)”在不同约定间可能对应不同 n、k、d。
Classical RM parameters are analytic, but quantum parameters also depend on nesting, puncturing, and subsystem extensions. Report the full construction tuple because “QRM(m)” may denote different n, k, and d across conventions.
15 量子比特 punctured QRM 的标志性性质是物理逐比特 T 实现逻辑 T（在相位约定修正下）。该性质属于特定三正交/多正交结构，不能概括为所有 QRM 均有相同横向门。
The hallmark of the 15-qubit punctured QRM code is a qubitwise physical T implementing logical T up to convention-dependent phases. This belongs to a specific triorthogonal or higher-orthogonality structure and is not shared by every QRM code.
可分别对底层经典 RM 码做递归多数逻辑、快速变换或最大似然近似，再处理 CSS 陪集。线路解码还需加入重复综合征和相关 hook error；经典解码成功不等于量子退化解码最优。
The underlying classical RM codes can be decoded by recursive majority logic, fast transforms, or likelihood approximations, followed by CSS coset handling. Circuit decoding must include repeated syndromes and hook correlations; classical decoding success is not automatically optimal quantum degenerate decoding.
[[15,1,3]] 常与 Steane/彩色码通过码切换或 code morphing 组合，以便在 Clifford 与 T 门之间选择有利码。切换电路、gauge 测量和中间纠错的故障必须计入，不能只比较横向层深。
The [[15,1,3]] code is often combined with Steane or color codes through code switching or morphing to access favorable Clifford and T gates. Faults in switching circuits, gauge measurements, and intermediate correction must be counted rather than comparing transversal layer depth alone.
有限实例适合演示编码 T、魔态蒸馏块和码切换，但 15 个数据量子比特之外还需要综合征与验证资源。实验应区分逻辑门过程保真度与包含纠错后的逻辑错误率。
Finite instances support demonstrations of encoded T, magic-state-distillation blocks, and code switching, but require syndrome and verification resources beyond fifteen data qubits. Experiments should separate logical-gate process fidelity from corrected logical error rate.
QRM 与经典 RM、彩色码维度层级和三正交码存在紧密联系；15 量子比特实例也常作为 Reed–Muller 魔态蒸馏协议的编码骨架。共享横向门不等同于稳定子码等价。
QRM codes relate closely to classical RM codes, dimensional color-code hierarchies, and triorthogonal codes; the fifteen-qubit instance also underlies Reed-Muller magic-state distillation. Sharing a transversal gate does not establish stabilizer-code equivalence.
精确外部映射至少需要 [[n,k,d]]、RM 阶数、puncture/shorten 规则、稳定子生成元和逻辑相位约定。参数相同的 CSS 码可能没有同一横向 T 实现。
Exact external mapping needs [[n,k,d]], RM order, puncture or shortening rule, stabilizer generators, and logical-phase convention. A CSS code with equal parameters need not have the same transversal T implementation.