Algebra of Bivariate-Bicycle Surface Codes

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.

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2606.08771 [quant-ph]

(or arXiv:2606.08771v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.08771 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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