Find a complete set of equivalence moves for two-block codes

Construct a complete set of equivalence moves that characterizes when two two-block group-algebra codes are equivalent, beyond the currently used translations, group automorphisms, antipode with block swap, and X/Z exchange.

Background

The search enumerates two-block codes using four known classes of transformations: independent two-sided translations of the blocks, simultaneous group automorphisms, antipode with block swap, and exchange of X and Z. These moves substantially reduce the search space but are not known to capture every equivalence.

The authors therefore apply a Tanner-graph canonical key as an additional merge step. Because this canonicalization is not a theoretical characterization of stabilizer-code equivalence, a complete equivalence theory remains unresolved.

References

We note that the equivalence moves are not complete -- to the best of our knowledge, a complete set of equivalence moves is an open question.

— Bicycle Race: small two-block codes on the Pareto frontier  (2610.13880 - Lee, 9 Oct 2026) in Appendix, Section “Enumeration up to equivalence”