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= , Dic3 = ); the group of the board entry 672-20-32. Element x^i y^j (i mod 12, j mod 28) has index e = 168 (j mod 2) + 14 i + floor(j/2); it is qubit e of block 1 and qubit 336 + e of block 2; product (x^i y^j)(x^k y^l) = x^(i + 5^j k) y^(j + l). H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with L(a)[c,g] = 1 iff c g^-1 in a, R(b)[c,h] = 1 iff h^-1 c in b: the X-check at c is {s^-1 c : s in a} in block 1 and {c t^-1 : t in b} in block 2, the Z-check at c is {c t : t in b} in block 1 and {s c : s in a} in block 2, checks in the order of the index of c. a = 1 + x^2 y^9 + x^10 y^11, b = 1 + x^10 y^22 + x^7 y^27 (indices 0, 200, 313 and 0, 151, 279). n = 672, k = 28, check weight 6 = 3 + 3, every qubit in 3 X-checks and 3 Z-checks. Found by a seeded random search over pairs (a, b) on 30 non-abelian groups of order 60 to 480, screened by information-set decoding; see the research note.", "references": [ "arXiv:2306.16400", "arXiv:2608.08996" ], "date": "2026-11-11", "model": "Claude Opus 5.5", "notes": "Exact distance: d_X = d_Z = 24. Every column of H_X and H_Z has weight 3, so every logical has even weight; two independently written exhaustive searches (exactA, exactB), anchored at the four qubit orbits of a Tanner-graph automorphism group of order 168, find no nontrivial logical of weight <= 22 on either side, and a third program, dist-m4ri 0.16.1, confirms this also without any symmetry; see the research note and certs/672-28-24.json (programs: github.com/TimoKellerMath/quantum-error-correction @ aaa18674c9a1, two-block-and-amplified-codes/src; records: github.com/TimoKellerMath/quantum-error-correction @ 2a15e549f6f4, pair-partition-and-two-block-codes/672-28-24/). Novelty: no CSS entry with n = 672 and k = 28 on the board (commit 82a52ee51) and no published code with these parameters known to us (targeted literature search of 2026-11-11); related work on the same group (Qian and Li, arXiv:2608.08996) is discussed in notes/672-28-24.md. The only entry with n = 672 and k = 28 is the stabilizer code 672-28-30 (check weight 24), a different code: it has stabilizer elements of Pauli weight 4, this code none below 6. Checked, not equivalent to an existing entry." }, "family": "2bga-coset" }