Generalizing fold-diagonal SS gates to larger binarization-and-concatenation codes

Develop a systematic method to construct fold-diagonal logical S_i S_j gates for larger members of the binarization-and-concatenation (B&C) phantom code family beyond the [[20,2,6]] instance derived from the [[5,1,3]]_4 qudit code, or establish constraints that prevent such generalization.

Background

The paper introduces a binarize-and-concatenate scheme that yields high-distance k=2 phantom codes from self-dual GF(4) qudit codes, and shows that for [20,2,6] fold-diagonal SS gates are found using solver techniques.

However, extending these fold-diagonal gates to larger codes in the same family proved elusive, leaving open whether a general construction exists or whether only isolated instances support such gates.