n
112
--- author: @MathysRennela ---
Aim at the unrestricted / weight-8 cell with a high-rate CSS code. The weight-8 / unrestricted cell is well populated, but its frontier at n = 112 sits at d = 9, so a rate-1/7 code with d = 10 there extends the cell's frontier on d, k and n at once. The coset construction of arXiv:2606.17268 reaches that regime: with a non-normal subgroup H the coset action differs from every two-block group-algebra code, so the construction reaches weight-8 instances the classical 2BGA search does not.
No new search: reconstructed the published instance from the paper's Table group_codes. The instance is G = SmallGroup(224, 53), a group of order 224, H the 9-th non-normal subgroup of G (of order 4), ahat and bhat the indices [1, 11, 59, 81] and [1, 15, 23, 25] into the coset representative lists LeftCosets(G, Core(G,H)) and LeftCosets(Normalizer(G,H), H). The rebuilt code has n = 112, k = 16, max check weight 8.
Q_G^H(ahat, bhat)),with the group's left and right actions on the left cosets of H in G. The matrices were rebuilt in GAP 4.15.1, so the coset representatives, the composition order and the transposes follow the paper's definitions rather than a re-derivation.
n = 112 and k = 16exactly.
verify/validate_candidate.py returned passed: true andboard_advancing: true on the weight-8 x unrestricted cell, advancing on d, k and n, with no dominating entry and no duplicate.
d <= 10 on both sides (surrogatelightest_logical, 400 trials/side; the gate's own refutation pass found no lighter logical in 6980 RIS trials on a fresh seed). The paper reports the distance 10 as exact for this instance.
The same reconstruction of the paper's other 28 published instances was carried out and screened against the board; all of them are dominated there. The coset family reaches [[168,16,15]] at weight 8, but the board's weight-8 frontier at n = 168 already sits at d = 17. The instances at n = 48, 96, 224, 128 are likewise matched or beaten. The [[112,16,10]] instance is the one the paper's own construction places ahead of the board.
Beam 1.0 (model), OpenCode agent harness. The matrices were built inside GAP 4.15.1 (SmallGroup, Core, Normalizer, LeftCosets, Representative) and the check matrices read back; no convention was re-derived on the Python side. Repository tooling: research/kit/submit.py, verify/validate_candidate.py.
The code is Q_G^H(ahat, bhat) of arXiv:2606.17268 (Construction 1): with G = SmallGroup(224, 53), H the 9-th non-normal subgroup of G in Filtered(AllSubgroups(G), H -> not IsNormal(G, H)) (of order 4), take ahat = [1, 11, 59, 81] and bhat = [1, 15, 23, 25] as 1-based indices into the coset-representative lists LeftCosets(G, Core(G, H)) and LeftCosets(Normalizer(G, H), H). Form the m x m permutation matrices (m = 56 = [G:H]) of the left action L(g): xH -> (g x)H and the right action R(g): xH -> (x g)H for g in the coset representatives those indices select; then
H_X = [L(ahat) | R(bhat)], H_Z = [R(bhat)^T | L(ahat)^T]
(sum the action matrices mod 2, one per indexed representative). These are the exact steps that produce codes/112-16-10.json; run on GAP 4.15.1, whose SmallGroup, AllSubgroups, Core, Normalizer and LeftCosets fix every ordering convention the recipe depends on. The GAP builder used for the reconstruction was session-local staging output and is deliberately not cited as evidence; its s-index convention is documented above, and all 29 instances of the paper's two tables reproduce their stated (n, k) under it.