Algebra of Bivariate-Bicycle Surface Codes

• URL: http://arxiv.org/abs/2606.08771v1
• Date: Sun, 07 Jun 2026 18:24:32 GMT
• Title: Algebra of Bivariate-Bicycle Surface Codes
• Abstract summary: We relate the properties of bi-bicycle-surface (BBS) codes to the number and location of their common roots.<n>The number of roots $(x,y)$ with finite, non-zero coordinates determines the multiplicity of the codes.<n>We formulate a prescription for constructing BBS codes that works for regions with rectangular, diagonal and tilted boundaries.
• Score: 1.66692958496368
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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