Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring ℤq[i]
Abstract
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=⟨α⟩, 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, C2/C1 parametrizes logical X-operator classes and Rqn/C2 parametrizes physical X-error syndrome classes. Using this syndrome quotient, we construct a transversal of C2 in Rqn 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.
Related
- Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
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BibTeX
@misc{tiwari2026hermitian,
title = {{Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring $\mathbbZ_q[i]$}},
author = {Akanksha Tiwari and Ritumoni Sarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14434},
eprint = {2609.14434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14434},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Hermitian Duality and Syndrome Structure of CSS-Type Codes over the ring ℤ_q[i]
AU - Akanksha Tiwari
AU - Ritumoni Sarma
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14434
UR - https://arxiv.org/abs/2609.14434
AB - 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=⟨α⟩, 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.
ER -