Predicting minimum-weight class counts from algebraic structure

Determine whether the numbers of minimum-weight logical classes in the annihilator and colon components, including counts such as the 36 one-sided annihilator classes appearing in three of the six standard bivariate bicycle codes, can be predicted from the structures of $ann(a)$ and $(a:b)$, potentially using classical defining-set or apparent-distance tools.

Background

The paper’s census enumerates all minimum-weight logical classes for six standard bivariate bicycle codes and records their algebraic component and representative shape. These counts vary substantially across codes, including the occurrence of 36 one-sided annihilator classes in three examples.

Although the annihilator quotient and colon quotient determine the algebraic decomposition of the logical space, the paper does not establish a formula that predicts the number or distribution of minimum-weight classes from these structures. It identifies classical defining-set and apparent-distance methods as possible tools for addressing the one-sided portion of this problem.

References

Two questions remain open. The first is an annihilator analogue of Corollary~\ref{cor:colonconn}, which would give $d_{\mathcal A}$ directly without passing through the full census. The second is whether the counts in Table~\ref{tab:census}, for instance the $36$ one-sided annihilator classes appearing in three of the codes, can be predicted from the structure of $ann(a)$ and $(a: b)$; classical defining-set and apparent-distance tools are natural candidates for the one-sided part.

— Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes  (2609.39395 - Rowshan et al., 30 Sep 2026) in Section 7.1, “Results and open problems” (Section~\ref{sec:disc-open})