Algebra of Bivariate-Bicycle Surface Codes

2026-06-07

AI Summary

This work proposes a unified construction method for bivariate bicycle surface (BBS) codes with arbitrary slanted boundaries. By analyzing the number and positions of common roots of the defining bivariate polynomials over extension fields, the study establishes an algebraic correspondence between these roots and both the code dimension and the geometric structure of the boundaries. The key insight lies in revealing how the coordinates of common roots map directly to boundary types—including slanted configurations—thereby enabling a consistent framework that eliminates the need for corner corrections. Leveraging tools from finite-field bivariate polynomial theory, monomial automorphisms of Laurent polynomial rings, and multiplicity-counting techniques from algebraic geometry, the method efficiently constructs BBS codes for rectangular, diagonal, and arbitrarily slanted boundaries while guaranteeing exact code dimensions and well-adapted boundary stabilizers.

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.

Problem

• bivariate-bicycle-surface codes
• common roots
• code dimension
• tilted boundaries
• monomial automorphisms

Innovation

• bivariate-bicycle-surface codes
• common roots
• monomial automorphisms
• tilted boundaries
• corner-free construction