n
48
Aim at the unrestricted / weight-6 cell with a high-rate CSS code. The weight-6 / unrestricted cell is thin at small n: the board's weight-6 frontier at n = 48 sits at d = 5, so a rate-1/6 code with d = 6 there extends the cell's frontier on d, k and n at once. The coset construction of arXiv:2606.17268 reaches exactly that regime: it produces weight-6 codes on groups where the classical 2BGA search of the same literature has nothing, because the subgroup H need not be normal and the coset action then differs from every two-block group-algebra code.
No new search: reconstructed the published instance from the paper's Table group_codes. The instance is G = SmallGroup(384, 512), a group of order 384, H the 53-rd non-normal subgroup of G (of order 16), ahat and bhat the indices [1, 34, 48] and [1, 6, 12] into the coset representative lists LeftCosets(G, Core(G,H)) and LeftCosets(Normalizer(G,H), H). The rebuilt code has n = 48, k = 8, max check weight 6.
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 = 48 and k = 8 exactly.verify/validate_candidate.py returned passed: true andboard_advancing: true on the weight-6 x unrestricted cell, advancing on d, k and n, with no dominating entry and no duplicate.
d <= 6 on both sides (surrogatelightest_logical, 400 trials/side; qldpc submit's accelerator pass re-searched at 2,000,000 trials per side and the gate's own refutation pass found no lighter logical in 4420 RIS trials on a fresh seed). The paper reports the distance 6 as exact for this instance.
codes/48-8-6.json, a different codewith the same [[48,8,6]] but max check weight 7; this one has max check weight 6, so it is filed as 48-8-6-b.
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 = 96, 224, 128 are likewise matched or beaten. The weight-6 instances at `n = 36, 48, 54, 56, 60, 72, 84, 90, 96, 108, 112, 120, 126, 144, 150, 168, 180, 186` are all dominated by existing weight-6 entries. The [[48,8,6]] 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(384, 512), H the 53-rd non-normal subgroup of G in Filtered(AllSubgroups(G), H -> not IsNormal(G, H)) (of order 16), take ahat = [1, 34, 48] and bhat = [1, 6, 12] 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 = 24 = [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/48-8-6-b.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.