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575, 653, 695, 716, 731 ], [ 478, 498, 553, 576, 654, 694, 715, 732 ], [ 489, 499, 577, 617, 655, 697, 705, 733 ], [ 479, 500, 578, 610, 656, 696, 704, 734 ], [ 487, 501, 579, 616, 657, 682, 702, 735 ], [ 498, 502, 566, 580, 639, 658, 729, 736 ], [ 500, 503, 561, 581, 627, 659, 722, 737 ], [ 484, 504, 559, 582, 626, 660, 721, 738 ], [ 495, 505, 549, 583, 661, 700, 708, 739 ], [ 485, 506, 546, 584, 662, 699, 707, 740 ], [ 493, 507, 548, 585, 663, 688, 703, 741 ], [ 504, 508, 572, 586, 645, 664, 735, 742 ], [ 506, 509, 567, 587, 630, 665, 728, 743 ], [ 490, 510, 565, 588, 629, 666, 727, 744 ], [ 501, 511, 552, 589, 632, 667, 714, 745 ], [ 491, 512, 547, 590, 631, 668, 713, 746 ], [ 499, 513, 551, 591, 624, 669, 706, 747 ], [ 510, 514, 578, 592, 651, 670, 741, 748 ], [ 512, 515, 573, 593, 636, 671, 734, 749 ], [ 496, 516, 571, 594, 635, 672, 733, 750 ], [ 507, 517, 558, 595, 638, 673, 720, 751 ], [ 497, 518, 550, 596, 637, 674, 719, 752 ], [ 505, 519, 557, 597, 625, 675, 712, 753 ], [ 516, 520, 584, 598, 657, 676, 747, 754 ], [ 518, 521, 579, 599, 642, 677, 740, 755 ], [ 502, 522, 577, 600, 641, 678, 739, 756 ], [ 513, 523, 564, 601, 644, 679, 726, 757 ], [ 503, 524, 556, 602, 643, 680, 725, 758 ], [ 511, 525, 563, 603, 628, 681, 718, 759 ], [ 522, 526, 590, 604, 663, 682, 753, 760 ], [ 524, 527, 585, 605, 648, 683, 746, 761 ], [ 508, 528, 583, 606, 647, 684, 745, 762 ], [ 519, 529, 570, 607, 650, 685, 732, 763 ], [ 509, 530, 562, 608, 649, 686, 731, 764 ], [ 517, 531, 569, 609, 634, 687, 724, 765 ], [ 528, 532, 596, 610, 669, 688, 759, 766 ], [ 530, 533, 591, 611, 654, 689, 752, 767 ], [ 514, 534, 589, 612, 653, 690, 751, 768 ], [ 525, 535, 576, 613, 656, 691, 738, 769 ], [ 515, 536, 568, 614, 655, 692, 737, 770 ], [ 523, 537, 575, 615, 640, 693, 730, 771 ], [ 536, 538, 597, 616, 660, 694, 758, 772 ], [ 520, 539, 595, 617, 659, 695, 757, 773 ], [ 531, 540, 582, 618, 662, 696, 744, 774 ], [ 521, 541, 574, 619, 661, 697, 743, 775 ], [ 529, 542, 581, 620, 646, 698, 736, 776 ], [ 537, 543, 588, 621, 668, 699, 750, 777 ], [ 527, 544, 580, 622, 667, 700, 749, 778 ] ] }, "distance": { "d": 6, "X": { "value": 6, "confidence": "upper_bound", "witness": [ 0, 12, 66, 94, 144, 181 ] }, "Z": { "value": 6, "confidence": "upper_bound", "witness": [ 0, 4, 6, 272, 305, 521 ] } }, "provenance": { "authors": [ "@hirobumii" ], "construction": "G=S3 x C13, GroupNames78,3. Reconstruct the first literal permutation generating set on39 points from https://people.maths.bris.ac.uk/~matyd/GroupNames/73/S3xC13.html (source SHA256 680d5f4ffd46681063d209bc2fa8538d254c6b8f6015b3559f4c95433986f453). One-based cycle generators are [[[1,2,3,4,5,6,7,8,9,10,11,12,13],[14,15,16,17,18,19,20,21,22,23,24,25,26],[27,28,29,30,31,32,33,34,35,36,37,38,39]],[[1,39,15],[2,27,16],[3,28,17],[4,29,18],[5,30,19],[6,31,20],[7,32,21],[8,33,22],[9,34,23],[10,35,24],[11,36,25],[12,37,26],[13,38,14]],[[14,38],[15,39],[16,27],[17,28],[18,29],[19,30],[20,31],[21,32],[22,33],[23,34],[24,35],[25,36],[26,37]]]. Convert to permutations, close from identity by appending products x*g in listed generator order, and index in first-discovery order; composition (x*g)[i]=x[g[i]]. L_g[t,u]=1 iff u=g*t; R_h[t,u]=1 iff u=t*h. A=[I+L6,I+L26,I+L33]; B=[I+R12,I+R38,I+R45], binary sums. Nine VV(k,j) blocks index3*k+j precede oneCC block, each size78. X(j) has A(k) onVV(k,j) and B(j) onCC. Z(k) has B(j)^T onVV(k,j) and A(k)^T onCC. Keep first independent original rows,231 per sector.", "references": [ "https://arxiv.org/abs/2111.03654", "https://arxiv.org/abs/2607.28795" ], "date": "2026-10-17", "notes": "Displayed presentation: maximum check weight 8, maximum qubit degree 12, 462 checks and total incidence 3696.", "model": "GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration)" }, "family": "lifted-product" }