Designing Group-Valued Codes with Full Regular Low-Weight Bases

Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width $f$, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance $d$ and check weight $w$. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with $f=22$ and an $[[1024,256,24]]$ PP code with $f\le 27$, both at $w=10$. At $w=11$, our PP constructions attain either $d=25$ with $f\le 27$ or $d=24$ with the proved optimum $f=25$, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.