{ "schema_version": "0.1", "name": "[[540,112,8]] non-abelian lifted product over ZSZ(18,6,7), check weight 6", "code_type": "CSS", "n": 540, "k": 112, "checks": { "X": [ [ 0, 13, 216, 298, 432, 513 ], [ 1, 14, 217, 299, 433, 442 ], [ 2, 15, 218, 294, 434, 479 ], [ 3, 16, 219, 295, 435, 510 ], [ 4, 17, 220, 296, 436, 439 ], [ 5, 12, 221, 297, 437, 476 ], [ 6, 55, 222, 232, 438, 519 ], [ 7, 56, 223, 233, 439, 448 ], [ 8, 57, 224, 228, 440, 485 ], [ 9, 58, 225, 229, 441, 516 ], [ 10, 59, 226, 230, 442, 445 ], [ 11, 54, 227, 231, 443, 482 ], [ 12, 97, 228, 274, 444, 525 ], [ 13, 98, 229, 275, 445, 454 ], [ 14, 99, 230, 270, 446, 491 ], [ 15, 100, 231, 271, 447, 522 ], [ 16, 101, 232, 272, 448, 451 ], [ 17, 96, 233, 273, 449, 488 ], [ 18, 31, 234, 316, 450, 531 ], [ 19, 32, 235, 317, 451, 460 ], [ 20, 33, 236, 312, 452, 497 ], [ 21, 34, 237, 313, 453, 528 ], [ 22, 35, 238, 314, 454, 457 ], [ 23, 30, 239, 315, 455, 494 ], [ 24, 73, 240, 250, 456, 537 ], [ 25, 74, 241, 251, 457, 466 ], [ 26, 75, 242, 246, 458, 503 ], [ 27, 76, 243, 247, 459, 534 ], [ 28, 77, 244, 248, 460, 463 ], [ 29, 72, 245, 249, 461, 500 ], [ 7, 30, 246, 292, 435, 462 ], [ 8, 31, 247, 293, 463, 472 ], [ 9, 32, 248, 288, 464, 509 ], [ 10, 33, 249, 289, 432, 465 ], [ 11, 34, 250, 290, 466, 469 ], [ 6, 35, 251, 291, 467, 506 ], [ 36, 49, 226, 252, 441, 468 ], [ 37, 50, 227, 253, 469, 478 ], [ 38, 51, 222, 254, 470, 515 ], [ 39, 52, 223, 255, 438, 471 ], [ 40, 53, 224, 256, 472, 475 ], [ 41, 48, 225, 257, 473, 512 ], [ 42, 91, 258, 268, 447, 474 ], [ 43, 92, 259, 269, 475, 484 ], [ 44, 93, 260, 264, 476, 521 ], [ 45, 94, 261, 265, 444, 477 ], [ 46, 95, 262, 266, 478, 481 ], [ 47, 90, 263, 267, 479, 518 ], [ 25, 48, 264, 310, 453, 480 ], [ 26, 49, 265, 311, 481, 490 ], [ 27, 50, 266, 306, 482, 527 ], [ 28, 51, 267, 307, 450, 483 ], [ 29, 52, 268, 308, 484, 487 ], [ 24, 53, 269, 309, 485, 524 ], [ 54, 67, 244, 270, 459, 486 ], [ 55, 68, 245, 271, 487, 496 ], [ 56, 69, 240, 272, 488, 533 ], [ 57, 70, 241, 273, 456, 489 ], [ 58, 71, 242, 274, 490, 493 ], [ 59, 66, 243, 275, 491, 530 ], [ 1, 60, 276, 286, 465, 492 ], [ 2, 61, 277, 287, 493, 502 ], [ 3, 62, 278, 282, 494, 539 ], [ 4, 63, 279, 283, 462, 495 ], [ 5, 64, 280, 284, 496, 499 ], [ 0, 65, 281, 285, 497, 536 ], [ 43, 66, 220, 282, 471, 498 ], [ 44, 67, 221, 283, 499, 508 ], [ 45, 68, 216, 284, 437, 500 ], [ 46, 69, 217, 285, 468, 501 ], [ 47, 70, 218, 286, 502, 505 ], [ 42, 71, 219, 287, 434, 503 ], [ 72, 85, 262, 288, 477, 504 ], [ 73, 86, 263, 289, 505, 514 ], [ 74, 87, 258, 290, 443, 506 ], [ 75, 88, 259, 291, 474, 507 ], [ 76, 89, 260, 292, 508, 511 ], [ 77, 84, 261, 293, 440, 509 ], [ 19, 78, 294, 304, 483, 510 ], [ 20, 79, 295, 305, 511, 520 ], [ 21, 80, 296, 300, 449, 512 ], [ 22, 81, 297, 301, 480, 513 ], [ 23, 82, 298, 302, 514, 517 ], [ 18, 83, 299, 303, 446, 515 ], [ 61, 84, 238, 300, 489, 516 ], [ 62, 85, 239, 301, 517, 526 ], [ 63, 86, 234, 302, 455, 518 ], [ 64, 87, 235, 303, 486, 519 ], [ 65, 88, 236, 304, 520, 523 ], [ 60, 89, 237, 305, 452, 521 ], [ 90, 103, 280, 306, 495, 522 ], [ 91, 104, 281, 307, 523, 532 ], [ 92, 105, 276, 308, 461, 524 ], [ 93, 106, 277, 309, 492, 525 ], [ 94, 107, 278, 310, 526, 529 ], [ 95, 102, 279, 311, 458, 527 ], [ 37, 96, 312, 322, 501, 528 ], [ 38, 97, 313, 323, 529, 538 ], [ 39, 98, 314, 318, 467, 530 ], [ 40, 99, 315, 319, 498, 531 ], [ 41, 100, 316, 320, 532, 535 ], [ 36, 101, 317, 321, 464, 533 ], [ 79, 102, 256, 318, 507, 534 ], [ 80, 103, 257, 319, 436, 535 ], [ 81, 104, 252, 320, 473, 536 ], [ 82, 105, 253, 321, 504, 537 ], [ 83, 106, 254, 322, 433, 538 ], [ 78, 107, 255, 323, 470, 539 ], [ 108, 121, 324, 406, 432, 449 ], [ 109, 122, 325, 407, 433, 516 ], [ 110, 123, 326, 402, 434, 481 ], [ 111, 124, 327, 403, 435, 446 ], [ 112, 125, 328, 404, 436, 519 ], [ 113, 120, 329, 405, 437, 484 ], [ 114, 163, 330, 340, 438, 455 ], [ 115, 164, 331, 341, 439, 522 ], [ 116, 165, 332, 336, 440, 487 ], [ 117, 166, 333, 337, 441, 452 ], [ 118, 167, 334, 338, 442, 525 ], [ 119, 162, 335, 339, 443, 490 ], [ 120, 205, 336, 382, 444, 461 ], [ 121, 206, 337, 383, 445, 528 ], [ 122, 207, 338, 378, 446, 493 ], [ 123, 208, 339, 379, 447, 458 ], [ 124, 209, 340, 380, 448, 531 ], [ 125, 204, 341, 381, 449, 496 ], [ 126, 139, 342, 424, 450, 467 ], [ 127, 140, 343, 425, 451, 534 ], [ 128, 141, 344, 420, 452, 499 ], [ 129, 142, 345, 421, 453, 464 ], [ 130, 143, 346, 422, 454, 537 ], [ 131, 138, 347, 423, 455, 502 ], [ 132, 181, 348, 358, 456, 473 ], [ 133, 182, 349, 359, 432, 457 ], [ 134, 183, 350, 354, 458, 505 ], [ 135, 184, 351, 355, 459, 470 ], [ 136, 185, 352, 356, 435, 460 ], [ 137, 180, 353, 357, 461, 508 ], [ 115, 138, 354, 400, 462, 479 ], [ 116, 139, 355, 401, 438, 463 ], [ 117, 140, 356, 396, 464, 511 ], [ 118, 141, 357, 397, 465, 476 ], [ 119, 142, 358, 398, 441, 466 ], [ 114, 143, 359, 399, 467, 514 ], [ 144, 157, 334, 360, 468, 485 ], [ 145, 158, 335, 361, 444, 469 ], [ 146, 159, 330, 362, 470, 517 ], [ 147, 160, 331, 363, 471, 482 ], [ 148, 161, 332, 364, 447, 472 ], [ 149, 156, 333, 365, 473, 520 ], [ 150, 199, 366, 376, 474, 491 ], [ 151, 200, 367, 377, 450, 475 ], [ 152, 201, 368, 372, 476, 523 ], [ 153, 202, 369, 373, 477, 488 ], [ 154, 203, 370, 374, 453, 478 ], [ 155, 198, 371, 375, 479, 526 ], [ 133, 156, 372, 418, 480, 497 ], [ 134, 157, 373, 419, 456, 481 ], [ 135, 158, 374, 414, 482, 529 ], [ 136, 159, 375, 415, 483, 494 ], [ 137, 160, 376, 416, 459, 484 ], [ 132, 161, 377, 417, 485, 532 ], [ 162, 175, 352, 378, 486, 503 ], [ 163, 176, 353, 379, 462, 487 ], [ 164, 177, 348, 380, 488, 535 ], [ 165, 178, 349, 381, 489, 500 ], [ 166, 179, 350, 382, 465, 490 ], [ 167, 174, 351, 383, 491, 538 ], [ 109, 168, 384, 394, 492, 509 ], [ 110, 169, 385, 395, 468, 493 ], [ 111, 170, 386, 390, 433, 494 ], [ 112, 171, 387, 391, 495, 506 ], [ 113, 172, 388, 392, 471, 496 ], [ 108, 173, 389, 393, 436, 497 ], [ 151, 174, 328, 390, 498, 515 ], [ 152, 175, 329, 391, 474, 499 ], [ 153, 176, 324, 392, 439, 500 ], [ 154, 177, 325, 393, 501, 512 ], [ 155, 178, 326, 394, 477, 502 ], [ 150, 179, 327, 395, 442, 503 ], [ 180, 193, 370, 396, 504, 521 ], [ 181, 194, 371, 397, 480, 505 ], [ 182, 195, 366, 398, 445, 506 ], [ 183, 196, 367, 399, 507, 518 ], [ 184, 197, 368, 400, 483, 508 ], [ 185, 192, 369, 401, 448, 509 ], [ 127, 186, 402, 412, 510, 527 ], [ 128, 187, 403, 413, 486, 511 ], [ 129, 188, 404, 408, 451, 512 ], [ 130, 189, 405, 409, 513, 524 ], [ 131, 190, 406, 410, 489, 514 ], [ 126, 191, 407, 411, 454, 515 ], [ 169, 192, 346, 408, 516, 533 ], [ 170, 193, 347, 409, 492, 517 ], [ 171, 194, 342, 410, 457, 518 ], [ 172, 195, 343, 411, 519, 530 ], [ 173, 196, 344, 412, 495, 520 ], [ 168, 197, 345, 413, 460, 521 ], [ 198, 211, 388, 414, 522, 539 ], [ 199, 212, 389, 415, 498, 523 ], [ 200, 213, 384, 416, 463, 524 ], [ 201, 214, 385, 417, 525, 536 ], [ 202, 215, 386, 418, 501, 526 ], [ 203, 210, 387, 419, 466, 527 ], [ 145, 204, 420, 430, 437, 528 ], [ 146, 205, 421, 431, 504, 529 ], [ 147, 206, 422, 426, 469, 530 ], [ 148, 207, 423, 427, 434, 531 ], [ 149, 208, 424, 428, 507, 532 ], [ 144, 209, 425, 429, 472, 533 ], [ 187, 210, 364, 426, 443, 534 ], [ 188, 211, 365, 427, 510, 535 ], [ 189, 212, 360, 428, 475, 536 ], [ 190, 213, 361, 429, 440, 537 ], [ 191, 214, 362, 430, 513, 538 ], [ 186, 215, 363, 431, 478, 539 ] ], "Z": [ [ 0, 33, 108, 133, 432, 497 ], [ 1, 106, 109, 170, 433, 492 ], [ 2, 71, 110, 207, 434, 493 ], [ 3, 30, 111, 136, 435, 494 ], [ 4, 103, 112, 173, 436, 495 ], [ 5, 68, 113, 204, 437, 496 ], [ 6, 39, 114, 139, 438, 467 ], [ 4, 7, 115, 176, 439, 462 ], [ 8, 77, 116, 213, 440, 463 ], [ 9, 36, 117, 142, 441, 464 ], [ 1, 10, 118, 179, 442, 465 ], [ 11, 74, 119, 210, 443, 466 ], [ 12, 45, 120, 145, 437, 444 ], [ 10, 13, 121, 182, 432, 445 ], [ 14, 83, 111, 122, 433, 446 ], [ 15, 42, 123, 148, 434, 447 ], [ 7, 16, 124, 185, 435, 448 ], [ 17, 80, 108, 125, 436, 449 ], [ 18, 51, 126, 151, 450, 515 ], [ 16, 19, 127, 188, 451, 510 ], [ 20, 89, 117, 128, 452, 511 ], [ 21, 48, 129, 154, 453, 512 ], [ 13, 22, 130, 191, 454, 513 ], [ 23, 86, 114, 131, 455, 514 ], [ 24, 57, 132, 157, 456, 485 ], [ 22, 25, 133, 194, 457, 480 ], [ 26, 95, 123, 134, 458, 481 ], [ 27, 54, 135, 160, 459, 482 ], [ 19, 28, 136, 197, 460, 483 ], [ 29, 92, 120, 137, 461, 484 ], [ 30, 63, 138, 163, 455, 462 ], [ 28, 31, 139, 200, 450, 463 ], [ 32, 101, 129, 140, 451, 464 ], [ 33, 60, 141, 166, 452, 465 ], [ 25, 34, 142, 203, 453, 466 ], [ 35, 98, 126, 143, 454, 467 ], [ 36, 69, 144, 169, 468, 533 ], [ 34, 37, 145, 206, 469, 528 ], [ 38, 107, 135, 146, 470, 529 ], [ 39, 66, 147, 172, 471, 530 ], [ 31, 40, 148, 209, 472, 531 ], [ 41, 104, 132, 149, 473, 532 ], [ 42, 75, 150, 175, 474, 503 ], [ 40, 43, 151, 212, 475, 498 ], [ 5, 44, 141, 152, 476, 499 ], [ 45, 72, 153, 178, 477, 500 ], [ 37, 46, 154, 215, 478, 501 ], [ 2, 47, 138, 155, 479, 502 ], [ 48, 81, 156, 181, 473, 480 ], [ 46, 49, 110, 157, 468, 481 ], [ 11, 50, 147, 158, 469, 482 ], [ 51, 78, 159, 184, 470, 483 ], [ 43, 52, 113, 160, 471, 484 ], [ 8, 53, 144, 161, 472, 485 ], [ 54, 87, 162, 187, 443, 486 ], [ 52, 55, 116, 163, 438, 487 ], [ 17, 56, 153, 164, 439, 488 ], [ 57, 84, 165, 190, 440, 489 ], [ 49, 58, 119, 166, 441, 490 ], [ 14, 59, 150, 167, 442, 491 ], [ 60, 93, 168, 193, 492, 521 ], [ 58, 61, 122, 169, 493, 516 ], [ 23, 62, 159, 170, 494, 517 ], [ 63, 90, 171, 196, 495, 518 ], [ 55, 64, 125, 172, 496, 519 ], [ 20, 65, 156, 173, 497, 520 ], [ 66, 99, 174, 199, 491, 498 ], [ 64, 67, 128, 175, 486, 499 ], [ 29, 68, 165, 176, 487, 500 ], [ 69, 96, 177, 202, 488, 501 ], [ 61, 70, 131, 178, 489, 502 ], [ 26, 71, 162, 179, 490, 503 ], [ 72, 105, 180, 205, 461, 504 ], [ 70, 73, 134, 181, 456, 505 ], [ 35, 74, 171, 182, 457, 506 ], [ 75, 102, 183, 208, 458, 507 ], [ 67, 76, 137, 184, 459, 508 ], [ 32, 77, 168, 185, 460, 509 ], [ 3, 78, 186, 211, 510, 539 ], [ 76, 79, 140, 187, 511, 534 ], [ 41, 80, 177, 188, 512, 535 ], [ 0, 81, 189, 214, 513, 536 ], [ 73, 82, 143, 190, 514, 537 ], [ 38, 83, 174, 191, 515, 538 ], [ 9, 84, 109, 192, 509, 516 ], [ 82, 85, 146, 193, 504, 517 ], [ 47, 86, 183, 194, 505, 518 ], [ 6, 87, 112, 195, 506, 519 ], [ 79, 88, 149, 196, 507, 520 ], [ 44, 89, 180, 197, 508, 521 ], [ 15, 90, 115, 198, 479, 522 ], [ 88, 91, 152, 199, 474, 523 ], [ 53, 92, 189, 200, 475, 524 ], [ 12, 93, 118, 201, 476, 525 ], [ 85, 94, 155, 202, 477, 526 ], [ 50, 95, 186, 203, 478, 527 ], [ 21, 96, 121, 204, 449, 528 ], [ 94, 97, 158, 205, 444, 529 ], [ 59, 98, 195, 206, 445, 530 ], [ 18, 99, 124, 207, 446, 531 ], [ 91, 100, 161, 208, 447, 532 ], [ 56, 101, 192, 209, 448, 533 ], [ 27, 102, 127, 210, 527, 534 ], [ 100, 103, 164, 211, 522, 535 ], [ 65, 104, 201, 212, 523, 536 ], [ 24, 105, 130, 213, 524, 537 ], [ 97, 106, 167, 214, 525, 538 ], [ 62, 107, 198, 215, 526, 539 ], [ 216, 249, 324, 349, 432, 500 ], [ 217, 322, 325, 386, 433, 501 ], [ 218, 287, 326, 423, 434, 502 ], [ 219, 246, 327, 352, 435, 503 ], [ 220, 319, 328, 389, 436, 498 ], [ 221, 284, 329, 420, 437, 499 ], [ 222, 255, 330, 355, 438, 470 ], [ 220, 223, 331, 392, 439, 471 ], [ 224, 293, 332, 429, 440, 472 ], [ 225, 252, 333, 358, 441, 473 ], [ 217, 226, 334, 395, 442, 468 ], [ 227, 290, 335, 426, 443, 469 ], [ 228, 261, 336, 361, 440, 444 ], [ 226, 229, 337, 398, 441, 445 ], [ 230, 299, 327, 338, 442, 446 ], [ 231, 258, 339, 364, 443, 447 ], [ 223, 232, 340, 401, 438, 448 ], [ 233, 296, 324, 341, 439, 449 ], [ 234, 267, 342, 367, 450, 518 ], [ 232, 235, 343, 404, 451, 519 ], [ 236, 305, 333, 344, 452, 520 ], [ 237, 264, 345, 370, 453, 521 ], [ 229, 238, 346, 407, 454, 516 ], [ 239, 302, 330, 347, 455, 517 ], [ 240, 273, 348, 373, 456, 488 ], [ 238, 241, 349, 410, 457, 489 ], [ 242, 311, 339, 350, 458, 490 ], [ 243, 270, 351, 376, 459, 491 ], [ 235, 244, 352, 413, 460, 486 ], [ 245, 308, 336, 353, 461, 487 ], [ 246, 279, 354, 379, 458, 462 ], [ 244, 247, 355, 416, 459, 463 ], [ 248, 317, 345, 356, 460, 464 ], [ 249, 276, 357, 382, 461, 465 ], [ 241, 250, 358, 419, 456, 466 ], [ 251, 314, 342, 359, 457, 467 ], [ 252, 285, 360, 385, 468, 536 ], [ 250, 253, 361, 422, 469, 537 ], [ 254, 323, 351, 362, 470, 538 ], [ 255, 282, 363, 388, 471, 539 ], [ 247, 256, 364, 425, 472, 534 ], [ 257, 320, 348, 365, 473, 535 ], [ 258, 291, 366, 391, 474, 506 ], [ 256, 259, 367, 428, 475, 507 ], [ 221, 260, 357, 368, 476, 508 ], [ 261, 288, 369, 394, 477, 509 ], [ 253, 262, 370, 431, 478, 504 ], [ 218, 263, 354, 371, 479, 505 ], [ 264, 297, 372, 397, 476, 480 ], [ 262, 265, 326, 373, 477, 481 ], [ 227, 266, 363, 374, 478, 482 ], [ 267, 294, 375, 400, 479, 483 ], [ 259, 268, 329, 376, 474, 484 ], [ 224, 269, 360, 377, 475, 485 ], [ 270, 303, 378, 403, 446, 486 ], [ 268, 271, 332, 379, 447, 487 ], [ 233, 272, 369, 380, 448, 488 ], [ 273, 300, 381, 406, 449, 489 ], [ 265, 274, 335, 382, 444, 490 ], [ 230, 275, 366, 383, 445, 491 ], [ 276, 309, 384, 409, 492, 524 ], [ 274, 277, 338, 385, 493, 525 ], [ 239, 278, 375, 386, 494, 526 ], [ 279, 306, 387, 412, 495, 527 ], [ 271, 280, 341, 388, 496, 522 ], [ 236, 281, 372, 389, 497, 523 ], [ 282, 315, 390, 415, 494, 498 ], [ 280, 283, 344, 391, 495, 499 ], [ 245, 284, 381, 392, 496, 500 ], [ 285, 312, 393, 418, 497, 501 ], [ 277, 286, 347, 394, 492, 502 ], [ 242, 287, 378, 395, 493, 503 ], [ 288, 321, 396, 421, 464, 504 ], [ 286, 289, 350, 397, 465, 505 ], [ 251, 290, 387, 398, 466, 506 ], [ 291, 318, 399, 424, 467, 507 ], [ 283, 292, 353, 400, 462, 508 ], [ 248, 293, 384, 401, 463, 509 ], [ 219, 294, 402, 427, 434, 510 ], [ 292, 295, 356, 403, 435, 511 ], [ 257, 296, 393, 404, 436, 512 ], [ 216, 297, 405, 430, 437, 513 ], [ 289, 298, 359, 406, 432, 514 ], [ 254, 299, 390, 407, 433, 515 ], [ 225, 300, 325, 408, 512, 516 ], [ 298, 301, 362, 409, 513, 517 ], [ 263, 302, 399, 410, 514, 518 ], [ 222, 303, 328, 411, 515, 519 ], [ 295, 304, 365, 412, 510, 520 ], [ 260, 305, 396, 413, 511, 521 ], [ 231, 306, 331, 414, 482, 522 ], [ 304, 307, 368, 415, 483, 523 ], [ 269, 308, 405, 416, 484, 524 ], [ 228, 309, 334, 417, 485, 525 ], [ 301, 310, 371, 418, 480, 526 ], [ 266, 311, 402, 419, 481, 527 ], [ 237, 312, 337, 420, 452, 528 ], [ 310, 313, 374, 421, 453, 529 ], [ 275, 314, 411, 422, 454, 530 ], [ 234, 315, 340, 423, 455, 531 ], [ 307, 316, 377, 424, 450, 532 ], [ 272, 317, 408, 425, 451, 533 ], [ 243, 318, 343, 426, 530, 534 ], [ 316, 319, 380, 427, 531, 535 ], [ 281, 320, 417, 428, 532, 536 ], [ 240, 321, 346, 429, 533, 537 ], [ 313, 322, 383, 430, 528, 538 ], [ 278, 323, 414, 431, 529, 539 ] ] }, "distance": { "d": 8, "X": { "value": 8, "confidence": "upper_bound", "witness": [ 233, 271, 288, 321, 332, 341, 369, 429 ] }, "Z": { "value": 8, "confidence": "upper_bound", "witness": [ 177, 199, 348, 358, 366, 389, 393, 398 ] } }, "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_18 x|_7 Z_6 (order 108; generators x, y with x^18 = y^6 = 1, y x = x^7 y; element x^a y^b has index a*6+b). Base matrices A = [1 + x^10y^5, 1 + x^11y^2] (entries act by the left regular representation L(g)[gh,h]=1) and B = [1 + x^13y^3, 1 + x^2y^5] (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 one fast RIS (gf2_fast) two-sided search of 300,000 trials per side (seed 7919) returned a weight-8 X logical; a separate deep confirmation of 1,000,000 trials per side (seed 20260916) found weight-8 logicals on both sides and nothing lighter (verdict corroborated). Gate refutation seed 612045197. Both binomial seed rows have Cayley-graph girth 8, which upper-bounds d; the code meets that bound. Novelty: no [[540,112,8]] 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(18,6,7)\", \"N\": 108, \"A\": [[[0, 65], [0, 68]]], \"B\": [[[0, 81], [0, 17]]], \"class_d\": [8, 8], \"w\": 6}" }, "family": "lifted-product" }