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299, 303, 330, 355, 366, 416, 432, 452, 463, 478 ] }, "Z": { "value": 26, "confidence": "exact", "witness": [ 16, 39, 54, 61, 105, 151, 177, 195, 201, 208, 232, 237, 245, 259, 283, 293, 336, 374, 384, 385, 416, 423, 429, 441, 451, 475 ] } }, "provenance": { "authors": [ "@TimoKellerMath" ], "origin": "submission", "novelty": "new_parameters", "construction": "Two-block group algebra (2BGA) code C(G;a,b) over G = C12 x Dic5 = x , |z| = 12, SmallGroup(240,40). Element z^c x^i y^j (c mod 12, i mod 5, j mod 4) has index 20c + 4i + j; it is qubit 20c + 4i + j of block 1 and 240 + 20c + 4i + j of block 2 (the indexing of 720-16-36 and 960-24-38); product (c,i,j)(c',i',j') = (c + c', i + (-1)^j i', j + j'). H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with L(a)[g,h] = 1 iff g = alpha h, R(b)[g,h] = 1 iff g = h beta: 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} and {s c : s in a}, checks in the order of the index of c. a = 1 + z^8 x^3 y^2 + z^11 y^3, b = 1 + z^2 x y^2 + y. n = 480, k = 16, check weight 6 = 3 + 3, every qubit in 3 X-checks and 3 Z-checks. Found by a design from the representation theory of F_2[G]: all 16 logical qubits lie in one block M_2(F_4[C4 x C2]) of F_2[G], where the norms of a and b meet in a curvilinear point of length 4.", "references": [ "arXiv:2306.16400", "arXiv:2305.06890", "arXiv:2610.13880", "arXiv:1611.07164", "arXiv:2611.08247" ], "date": "2026-11-10", "model": "Claude Opus 5.5", "notes": "Exact distance: d_X = d_Z = 26. 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 orbits of a Tanner-graph automorphism group of order 240, find no nontrivial logical of weight <= 24 on either side, in two sets of runs on two instances with independently computed groups; see the research note and certs/480-16-26.json (programs, completeness proofs, instances and logs: github.com/TimoKellerMath/quantum-error-correction @ aaa18674c9a1, two-block-and-amplified-codes/). tropidist b03ec2d4077f reached only d_X, d_Z >= 22 in 1200 s per side. Novelty: no [[480,16]] entry on the board (commit 8042f9da) and no published code with these parameters known to us (literature checked: see the note). The census of weight-6 2BGA codes over C_m x Dic5 reported in notes/720-16-36.md (group orders up to 480) very likely contains this code; it is not submitted or described there. No entry has these n and k; checked, not equivalent to an existing entry." }, "family": "2bga-coset" }