We relate the properties of bivariate-bicycle-surface (BBS) codes, constructed from a pair of bivariate polynomials over a finite field, to the number and location of their common roots in the extension field. The number of roots $(x,y)$ with finite, non-zero coordinates -- counted with algebraic multiplicity -- determines the dimension of the codes. This dimension is invariant under monomial automorphisms of the Laurent polynomial ring. Conversely, roots with zero or infinite $x$- or $y$-coordinates indicate that specialized generators are required near the corresponding boundary (e.g., the left or right boundary for a root where $x$ is zero or infinite, respectively). These roots can appear or disappear under monomial transformations, which reveals the structure of tilted boundaries. Based on these results, we formulate a prescription for constructing BBS codes that works for regions with rectangular, diagonal, and arbitrarily tilted boundaries. A key advantage of this approach is that no corner corrections are needed, provided the polynomials satisfy orientation-specific edge conditions.

翻译：我们建立了由有限域上一对二元多项式构造的双变量自行车表面码（BBS码）性质与其在扩域中公共根数目及位置之间的关系。具有有限非零坐标的根 $(x,y)$ ——按代数重数计数——的数目决定了码的维数。该维数在洛朗多项式环的单项式自同构下保持不变。反之，具有零或无穷 $x$ 或 $y$ 坐标的根表明，在相应边界附近需要专门的生成元（例如，对于 $x$ 为零或无穷的根，分别对应左或右边界）。这些根在单项式变换下可能出现或消失，从而揭示倾斜边界的结构。基于这些结果，我们提出了一种对具有矩形、对角线和任意倾斜边界区域均适用的BBS码构造方法。该方法的关键优势在于，只要多项式满足特定方向的边界条件，就无需进行角点修正。