拟循环码的零化子与扭曲欧几里得对偶
Annihilator and twisted Euclidean duality for quasi-polycyclic codes
- SRM University-AP（SRM大学安得拉邦分校）
- University of Valladolid（巴利亚多利德大学）
- Carleton University（卡尔顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究拟循环码的零化子对偶，证明其非退化且保持拟循环性，给出扭曲欧几里得描述及自正交判据，并应用于量子码构造，得到达到已知最优界的稳定子码。
AI中文摘要：
设 $f\in\mathbb F_q[x]$ 是次数为 $m$ 的首一多项式且 $f(0)\ne 0$，令 $\mathcal R=\mathbb F_q[x]/\langle f\rangle$。在系数展开下，指标为 $n$ 的拟循环（QP）码对应于 $\mathcal R^n$ 的一个 $\mathcal R$-子模。本文研究关于零化子对偶的 QP 码。我们证明该形式是非退化的，且 QP 码的零化子对偶仍是 QP 码。我们还给出了该对偶在 $\mathcal R$-值点积下的等价描述，由此得到自正交性判据。我们确定了零化子形式的 Gram 矩阵，并得到其行列式的显式公式。在系数坐标下，这表明零化子对偶可视为扭曲欧几里得对偶。利用这一描述，我们刻画了何时一个逐坐标的 $\mathbb F_q$-线性映射将零化子对偶转化为普通欧几里得对偶。对于无平方因子 $f$，我们证明零化子对偶在由中国剩余定理产生的分量上分解为普通欧几里得对偶。这给出了自正交、自对偶、含对偶和互补对偶 QP 码的简单判据。我们展示了零化子对偶与汉明重量枚举器的相互作用，并计算了与该对偶相关的 MacWilliams 变换。最后，我们将这些结果应用于 $\mathcal R$ 上的 Calderbank--Shor--Steane 和 Steane 扩展量子码构造，并在存在合适的保对偶坐标映射时应用于 $\mathbb F_q$ 上。这给出了二元和三元稳定子码，其最小距离下界与已知最佳界相匹配，其中大多数来自非域的环 $\mathcal R$。
英文摘要：
Let $f\in\mathbb F_q[x]$ be a monic polynomial of degree $m$ with $f(0)\ne 0$, and let $\mathcal R=\mathbb F_q[x]/\langle f\rangle$. Under coefficient expansion, a quasi-polycyclic (QP) code of index $n$ corresponds to an $\mathcal R$-submodule of $\mathcal R^n$. In this paper, we study QP codes with respect to the annihilator duality. We show that this form is non-degenerate and that the annihilator dual of a QP code is again a QP code. We also give an equivalent description of the dual in terms of the $\mathcal R$-valued dot product, which leads to self-orthogonality criteria. We determine the Gram matrix of the annihilator form and obtain an explicit formula for its determinant. In coefficient coordinates, this shows that the annihilator dual can be viewed as a twisted Euclidean dual. Using this description, we characterize when a coordinatewise $\mathbb F_q$-linear map converts annihilator duality into ordinary Euclidean duality. For squarefree $f$, we show that annihilator duality decomposes into ordinary Euclidean duality on the components arising from the Chinese Remainder Theorem. This gives simple criteria for self-orthogonal, self-dual, dual-containing, and complementary-dual QP codes. We show how the annihilator dual interacts with the Hamming weight enumerator and compute the MacWilliams transform associated with that duality. Finally, we apply these results to Calderbank--Shor--Steane and Steane-enlarged quantum-code constructions over $\mathcal R$ and, when a suitable duality-preserving coordinate map exists, over $\mathbb F_q$. This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings $\mathcal R$ that are not fields.
