{ "schema_version": "0.2", "name": "[[514,162,29]] prime-index cyclic generalized-bicycle code on Z_257", "code_type": "CSS", "n": 514, "k": 162, "checks": { "X": [ [ 17, 44, 46, 94, 142, 190, 192, 219, 242, 245, 248, 251, 262, 275, 283, 340, 354, 391, 405, 462, 470, 483, 495, 507 ], [ 18, 45, 47, 95, 143, 191, 193, 220, 243, 246, 249, 252, 263, 276, 284, 341, 355, 392, 406, 463, 471, 484, 496, 508 ], [ 19, 46, 48, 96, 144, 192, 194, 221, 244, 247, 250, 253, 264, 277, 285, 342, 356, 393, 407, 464, 472, 485, 497, 509 ], [ 20, 47, 49, 97, 145, 193, 195, 222, 245, 248, 251, 254, 265, 278, 286, 343, 357, 394, 408, 465, 473, 486, 498, 510 ], [ 21, 48, 50, 98, 146, 194, 196, 223, 246, 249, 252, 255, 266, 279, 287, 344, 358, 395, 409, 466, 474, 487, 499, 511 ], [ 22, 49, 51, 99, 147, 195, 197, 224, 247, 250, 253, 256, 267, 280, 288, 345, 359, 396, 410, 467, 475, 488, 500, 512 ], [ 0, 23, 50, 52, 100, 148, 196, 198, 225, 248, 251, 254, 268, 281, 289, 346, 360, 397, 411, 468, 476, 489, 501, 513 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"ris_gpu", "seeds": [ 88000007 ] } } }, "family": "generalized-bicycle", "provenance": { "authors": [ "@vprusso" ], "construction": "Generalized bicycle (two-block cyclic) code on Z_257 with m=257 PRIME. H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 514 qubits. a and b are low-weight codewords of a cyclic divisor ideal , g | x^257 - 1, assembled from a union of 2-cyclotomic cosets of Z_257 of total degree 80; k = 2 deg gcd(a, b, x^257 - 1) = 162. a=[17, 44, 46, 94, 142, 190, 192, 219, 242, 245, 248, 251], b=[5, 18, 26, 83, 97, 134, 148, 205, 213, 226, 238, 250]. Max check weight 24.", "references": [ "arXiv:2308.07915" ], "date": "2026-08-11", "notes": "Found by a search over PRIME block sizes m with 2m <= 700, the regime the [[482,146,42]] entry on Z_241 occupies; earlier cyclic searches here used composite m only. Distance ladder: 4k screen d<=48, 200k d<=50, 2M d<=46, 20M d<=41, each rung the minimum over its budget. Witnesses come from multi-seed randomized information-set searches at 20M reads and are re-verified to commute with the opposite check matrix and to lie outside the stabilizer rowspace. d is the minimum over sides (X 41, Z 44). Distance revised 2026-08-19 by GPU deep refutation with the board's standalone ris_gpu tool (verify/ris_gpu.cu, sqetch kernels by Muzhou Ma, MIT) on an NVIDIA L40S: a 1,000,000,000-trial random-ISD recovery ladder per side (seed 88000007) committed a weight-29 X logical and a weight-29 Z logical, both re-verified on the CPU with the repo's GF(2) arithmetic and recorded here as the new per-side witnesses. This tightens d<=41 to d<=29 and the score from 162*41^2/514 = 529.83 to 162*29^2/514 = 265.06. The distance remains a witness-backed upper bound, not an exact-distance claim. @vprusso constructed the code. Refutation credit and the search budget are recorded in witness_provenance on each corrected side (schema 0.2, issue #611); authorship is unchanged.", "model": "Claude Opus 5" } }