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113, 114, 123, 131, 156, 159, 165, 175, 184, 192, 217, 220, 226, 245, 264, 281, 349, 398, 407, 413, 430, 432, 449, 490, 547, 566, 579, 588, 596, 605, 613 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-25", "found_at_samples": 300000000, "survived_samples": 300000000, "tool": "verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified)", "seeds": [ 4102 ] } }, "Z": { "value": 36, "confidence": "upper_bound", "witness": [ 11, 35, 58, 63, 82, 104, 111, 121, 145, 185, 192, 199, 216, 228, 245, 262, 302, 309, 333, 347, 352, 378, 394, 436, 439, 446, 455, 465, 491, 516, 549, 554, 568, 604, 610, 615 ] } }, "provenance": { "authors": [ "@vprusso" ], "construction": "Generalized bicycle (two-block) code on the circulant ring Z_315: H_X=[A|B], H_Z=[B^T|A^T] with A, B the 315x315 circulants of a(x)=1+x^17+x^134 and b(x)=1+x^202+x^254, both weight 3 so the check weight is 6. k = 2 deg gcd(a,b,x^315-1) = 6: the shared factor is an irreducible cubic, which requires 7 | 315 and both polynomials divisible by the same cubic, a condition on the exponents mod 7. Found by generating that residue class directly rather than filtering for it. This k sits below the k>=8 filter of the exhaustive weight-6 trinomial enumeration over Z_N, N=181..350 recorded in the provenance of codes/630-12-34.json, so it lies outside that search.", "origin": "submission", "date": "2026-09-06", "model": "Claude Opus 5" }, "family": "generalized-bicycle" }