Directional codes, recently introduced by Gehér--Byfield--Ruban \cite{Geher2025Directional}, constitute a hardware-motivated family of quantum low-density parity-check (qLDPC) codes. These codes are defined by stabilizers measured by ancilla qubits executing a fixed \emph{direction word} (route) on square- or hex-grid connectivity. In this work, we develop a comprehensive \emph{word-first} analysis framework for route-generated, translation-invariant CSS codes on rectangular tori. Under this framework, a direction word $W$ deterministically induces a finite support pattern $P(W)$, from which we analytically derive: (i)~a closed-form route-to-support map; (ii)~the odd-multiplicity difference lattice $L(W)$ that classifies commutation-compatible $X/Z$ layouts; and (iii)~conservative finite-torus admissibility criteria. Furthermore, we provide: (iv)~a rigorous word equivalence and canonicalization theory (incorporating dihedral lattice symmetries, reversal/inversion, and cyclic shifts) to enable symmetry-quotiented searches; (v)~an ``inverse problem'' criterion to determine when a translation-invariant support pattern is realizable by a single route, including reconstruction and non-realizability certificates; and (vi)~a quasi-cyclic (group-algebra) reduction for row-periodic layouts that explains the sensitivity of code dimension $k$ to boundary conditions. As a case study, we analyze the word $W=\texttt{NE$^2$NE$^2$N}$ end-to-end. We provide explicit stabilizer dependencies, commuting-operator motifs, and an exact criterion for dimension collapse on thin rectangles: for $(L_x, L_y) = (2d, d)$ with row alternation, we find $k=4$ if $6 \mid d$, and $k=0$ otherwise.

翻译：方向性码是Gehér–Byfield–Ruban近期提出的一种受硬件启发的量子低密度奇偶校验码族。这类码由辅助量子位执行固定的方向字在方形或六边形网格连接上测量稳定子所定义。本文针对矩形环面上由路径生成的平移不变CSS码，建立了一套全面的字优先分析框架。在此框架下，方向字$W$确定性地诱导出一个有限支撑模式$P(W)$，并由此解析推导出：(i) 封闭形式的路径到支撑映射；(ii) 用于分类对易相容$X/Z$布局的奇重数差格$L(W)$；(iii) 保守的有限环面可容许性判据。此外，我们还提供了：(iv) 严格的字等价性与规范化理论（包含二面体格对称性、反转/逆序及循环移位），以支持对称性商空间的搜索；(v) 判定平移不变支撑模式是否可由单一路径实现的逆问题判据，包括重构方法与不可实现性证明；(vi) 针对行周期布局的拟循环（群代数）约化方法，用于解释码维$k$对边界条件的敏感性。作为案例研究，我们对字$W=\texttt{NE$^2$NE$^2$N}$进行了端到端分析。我们给出了显式的稳定子依赖关系、对易算子模式，以及窄矩形上维度坍缩的精确判据：对于采用行交替的$(L_x, L_y) = (2d, d)$结构，当$6 \mid d$时$k=4$，否则$k=0$。