Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring $\mathbb{Z}_q[i]$

RESUMO

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 $C=\langle\alpha\rangle$, we show that its Hermitian dual is $ C^{\perp_H}=\operatorname{Ann}(\sigma(\alpha)). $ 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.