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371, 380, 385, 387 ], [ 50, 68, 70, 105, 146, 157, 166, 193, 241, 277, 295, 333, 372, 381, 386, 388 ], [ 51, 69, 71, 106, 147, 158, 167, 194, 242, 278, 296, 334, 373, 382, 387, 389 ] ] }, "distance": { "d": 31, "X": { "value": 31, "confidence": "upper_bound", "witness": [ 9, 46, 61, 111, 118, 122, 136, 154, 163, 165, 168, 170, 188, 208, 210, 215, 224, 231, 236, 240, 252, 256, 263, 281, 311, 319, 324, 329, 349, 355, 374 ] }, "Z": { "value": 31, "confidence": "upper_bound", "witness": [ 16, 35, 41, 61, 66, 71, 79, 109, 127, 134, 138, 150, 154, 159, 166, 175, 180, 182, 202, 220, 222, 225, 227, 236, 254, 268, 272, 279, 329, 344, 381 ] } }, "provenance": { "authors": [ "@seunomonije" ], "model": "Claude Fable 5", "construction": "Generalized bicycle (two-block) code on the cyclic group Z_195: H_X = [A|B], H_Z = [B^T|A^T] with A, B circulants of a(x) = sum x^i, i in {0,2,7,16,55,93,111,147} (weight 8) and b(x) = sum x^i, i in {0,27,36,47,88,123,125,143} (weight 8); max check weight 16. Divisor trick: both a and b are weight-8 codewords of the classical cyclic code for g = gcd(a, b, x^195-1) of degree 41, giving k = 2*deg g = 82 by construction (machine-checked by rank). The weight-<=8 codeword pool of saturates at 12 cyclic-shift classes (60k+ RIS trials, 3 seeds); this is pool-class pair (0,1), one of the 51 previously-unexplored pairs from the exhaustive 66-pair enumeration of that pool. Distance evidence: randomized-information-set screen read d<=43 at 12k trials; staged deep ladder with fresh seed per rung (pair_depth 8) read 35 at 2M, 34 at 8M, 32 at 24M trials; final five-config barrage read [31, 32, 33, 36, 35] = three 8M-trial passes at pair_depth 8 with fresh seeds 777/999/424242 plus pair_depth-16 and pair_depth-24 passes at 3M trials. The lightest logical ever witnessed is the weight-31 Z logical from the seed-777 pass (support embedded below); the GB transpose symmetry (block swap + per-block index reversal i -> -i mod 195) maps it to the embedded weight-31 X logical, machine-verified against ker/rowspace. Claimed d <= 31 = the lightest witnessed logical.", "references": [ "arXiv:1904.02703", "arXiv:2203.17216" ], "date": "2026-07-18", "notes": "Search and verification by Claude Fable 5. Distance is an honest upper bound from ~30M randomized-information-set trials across 8 independent seeded passes and three pair depths; no logical below weight 31 was ever witnessed, but d could be lower. Literature novelty unverified. kd^2/n = 202.06; supersedes the [[390,82,30]] entry from the same divisor pool." }, "family": "generalized-bicycle" }