Two-block group-algebra codes

2BGA codes combine the two-block CSS ansatz with a finite group algebra. General non-Abelian groups make the family strictly broader than cyclic GB and two-factor Abelian BB. The essential structure is commuting left and right regular actions, not merely two square matrices.

For $a,b\in\mathbb F_q[G]$, let $A=L(a)$ and be left and right regular matrices. Then and with . A finite record must preserve group multiplication or a permutation representation, basis ordering, coefficients of , and block conventions.

Group automorphisms, left/right multiplication, and block swaps generate physical permutations and may induce logical Clifford actions. Nontrivial logical behavior depends on $a,b$, the group, and the logical basis, and requires an explicit symplectic-action calculation.

Expanded Tanner graphs can be decoded by BP, BP+OSD, or structure-aware methods, while short cycles, degeneracy, and group orbits affect message passing. The QECDB 2BGA filter supplies finite matrices, not a noise model, OSD order, stopping rule, or failure criterion.

The two-block form and bounded $W$ describe static checks only. Syndrome extraction must map left/right Cayley edges to hardware, color conflicts, control correlated faults, and evaluate circuit distance separately. QECirc currently has no 2BGA family page.

A finite identity should use the tuple (field, group presentation, ordered basis, a, b, block convention) plus hashes of normalized $H_X,$. Parameters or a group name alone do not remove duplicates from automorphisms, left/right multiplication, block swaps, and transposition.

The current QECDB 2BGA family filter returns 3820 records and is an appropriate family-level entry. Selecting a finite member still requires a permanent ID and normalized-matrix binding. GB, BB, and HGP labels are not interchangeable by parameters alone.