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295, 402, 516, 537, 633, 672 ], [ 296, 403, 472, 538, 634, 673 ], [ 294, 404, 428, 539, 635, 674 ] ] }, "distance": { "d": 8, "X": { "value": 8, "confidence": "upper_bound", "witness": [ 283, 345, 352, 369, 459, 479, 487, 504 ] }, "Z": { "value": 8, "confidence": "upper_bound", "witness": [ 41, 47, 79, 118, 272, 282, 311, 388 ] } }, "provenance": { "authors": [ "@vprusso" ], "origin": "submission", "novelty": "new_parameters", "model": "Claude Fable 5.1 (Claude Code)", "construction": "Lifted product over the non-abelian group algebra F_2[G], G = Z_45 x|_16 Z_3 (order 135; generators x, y with x^45 = y^3 = 1, y x = x^16 y; element x^a y^b has index a*3+b). Base matrices A = [1 + x^21y^1, 1 + x^32] (entries act by the left regular representation L(g)[gh,h]=1) and B = [1 + x^36y^2, 1 + x^7] (entries act by the right regular representation R(g)[h,hg]=1). Qubit blocks of size |G|: sector 1 holds (i,j) for i in cols(A), j in cols(B) at block i*n_B+j; sector 2 holds (r,s) for r in rows(A), s in rows(B). X-check (r,j) = [L(A[r][i]) on (i,j)] + [R(B[s][j]) on (r,s)]; Z-check (i,s) = [R(B[s][j])^T on (i,j)] + [L(A[r][i])^T on (r,s)]. Same construction as the weight-9 mitten / ZSZ-LP codes of arXiv:2607.28795 and arXiv:2607.27644 (which are the case of four weight-3 entries), here with entry weights [[2, 2]] / [[2, 2]].", "references": [ "arXiv:2607.28795", "arXiv:2607.27644", "arXiv:2111.03654" ], "date": "2026-09-16", "notes": "Distance is a witnessed upper bound (both sides weight 8). Depth: screen at 300 fast RIS trials read d <= 8, ladder 10k and 100k trials read 8, 8; at packaging two independent fast RIS (gf2_fast) two-sided searches of 1,000,000 trials per side (seeds 7919 and 15838) each returned a weight-8 X logical and nothing lighter on either side. Gate refutation seed 375696323. Both binomial seed rows have Cayley-graph girth 8, which upper-bounds d; the code meets that bound. Novelty: no [[675,139,8]] and no isomorphic code found in the 2BGA, GB, BB, QECDB and codetables data (Tanner-graph canonical-form check with pynauty); parameters claimed new, not verifier-proved. Not equivalent to a board entry (validator dedup found no exact or WL-equivalent match). Sampler spec: {\"family\": \"nonabelian-lp\", \"group\": \"ZSZ(45,3,16)\", \"N\": 135, \"A\": [[[0, 64], [0, 96]]], \"B\": [[[0, 110], [0, 21]]], \"class_d\": [8, 8], \"w\": 6}" }, "family": "lifted-product" }