环 $\mathbb{Z}_q[i]$ 上CSS型码的Hermitian对偶与综合征结构

中文摘要 AI 辅助

我们研究了环 $ R_q=\mathbb Z_q[i]$ 上的主理想码和CSS型稳定子构造的代数结构，其中 $q$ 的每个素因子都同余于 $1$ 模 $4$。尽管此类基于环的构造近来已在经典和量子纠错背景下被考虑，但主理想基数、Hermitian对偶、代数陪集和稳定子综合征之间的精确关系仍需进一步的结构分析。我们得到了主理想及其零化子的基数公式。对于长度为一的主理想码 $C=\langle\alpha\rangle$，我们证明其Hermitian对偶为 $ C^{\perp_H}=\operatorname{Ann}(\sigma(\alpha)). $ 对于满足 $ C_2^{\perp_H}\subseteq C_1\subseteq C_2 $ 的嵌套码，我们确定了相应的CSS型稳定子，并证明了物理 $X$ 错误综合征映射的核为 $ \ker(\operatorname{Syn}_X)=C_2. $ 因此，$C_2/C_1$ 参数化逻辑 $X$ 算子类，而 $R_q^n/C_2$ 参数化物理 $X$ 错误综合征类。利用该综合征商，我们构造了 $C_2$ 在 $R_q^n$ 中的一个横截，其元素具有两两不同的物理 $X$ 错误综合征。相应的 $X$ 型错误族可通过稳定子综合征测量进行校正，从而产生一种与稳定子结构一致的基于综合征的恢复过程。显式例子说明了由此产生的对偶、基数和综合征结构。

英文摘要

We study the algebraic structure of principal ideal codes and CSS-type stabilizer constructions over the ring $ R_q=\mathbb Z_q[i],$ where every prime divisor of $q$ is congruent to $1$ modulo $4$. Although such ring-based constructions have recently been considered in the context of classical and quantum error correction, the precise relationship among principal ideal cardinalities, Hermitian duality, algebraic cosets, and stabilizer syndromes requires further structural analysis. We obtain cardinality formulas for principal ideals and their annihilators. For a length-one principal ideal code $C=\langleα\rangle$, we show that its Hermitian dual is $ C^{\perp_H}=\operatorname{Ann}(σ(α)). $ For nested codes satisfying $ C_2^{\perp_H}\subseteq C_1\subseteq C_2, $ we determine the corresponding CSS-type stabilizer and prove that the kernel of the physical $X$-error syndrome map is $ \ker(\operatorname{Syn}_X)=C_2. $ Consequently, $C_2/C_1$ parametrizes logical $X$-operator classes and $R_q^n/C_2$ parametrizes physical $X$-error syndrome classes. Using this syndrome quotient, we construct a transversal of $C_2$ in $R_q^n$ whose elements have pairwise distinct physical $X$-error syndromes. The corresponding $X$-type error family is correctable by stabilizer syndrome measurement, yielding a syndrome-based recovery procedure that is consistent with the stabilizer structure. Explicit examples illustrate the resulting duality, cardinality, and syndrome structure.

发表机构

• Indian Institute of Technology Delhi（印度德里理工学院）

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