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[[560,2,22]] d ≤
n
560
k
2
d
22
kd²/n
1.729
w
4
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[7, 44, 57, 109, 112, 122, 134, 222, 224, 235, 259, 277, 324, 349, 354, 390, 403, 414, 430, 437, 450, 510]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[26, 89, 106, 164, 171, 182, 208, 213, 262, 273, 280, 345, 376, 394, 427, 482, 488, 513, 523, 537, 554, 559]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×560 (2,2)×1680 (3,2)×5040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 560 (2,2): 1680 (3,2): 5040 (3,4): 1120
trapping sets H_Z (1,2)×560 (2,2)×1680 (3,2)×5040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 560 (2,2): 1680 (3,2): 5040 (3,4): 1120

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Quadricycle code (rank-4 generalized bicycle on the abelian torus G = Z_2 x Z_5 x Z_7 x Z_4): H_X = [A | B], H_Z = [B^T | A^T] with A = x22 x34 x43 + x1 x23 x3 x43 and B = x1 x33 x4 + x32 x42, monomials of the four independent cyclic shifts over F_2; check weight 4. Found by uniform random sampling of weight-2+2 monomial sets (1,200 candidates, dims in [2,11] with prod <= 350) screened with a staged RIS funnel (300 -> 3,000 -> 20,000 trials, gf2_fast) and confirmed with a 1M-sample adversarial ladder. Generalizes the bivariate bicycles (arXiv:2308.07915) and the trivariate codes (arXiv:2406.19151) to four independent shifts.
model GLM 5.3 Flash (Zed agent) (claimed, not verified)
date 2026-09-20
notes Checked against the current board by the validation gate: verifies, not refuted, and no exact or WL-equivalent duplicate of any existing entry; not known to be equivalent to any existing entry under a code symmetry.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[560,2,22]] — quadricycle code on Z_2 × Z_5 × Z_7 × Z_4

Direction & hypothesis

Track cell weight-4 × unrestricted (record was d ≤ 21). Quadricycle = rank-4 generalized bicycle: A, B over F_2[Z_l1 × Z_l2 × Z_l3 × Z_l4], H_X = [A | B], H_Z = [Bᵀ | Aᵀ], four independent cyclic shifts (unlike the trivariate codes of arXiv:2406.19151, which reduce to rank 2). Same pilot sweep as [[700,2,25]]; this is the smallest-n record-beater of the three, so the best efficiency placement of them: kd²/n = 1.93, essentially tying the [[454,2,21]] 2BGA it advances on distance.

What was searched

1,200 weight-2+2 quadricycles: dims uniform in [2, 11] with prod ≤ 350, two distinct monomials per side uniform over the exponent 4-tuples, seed 101. Staged RIS funnel 300 → 3,000 → 20,000 trials (gf2_fast, 8 threads; kit screen_adaptive in research/kit/search.py, surrogate in research/kit/surrogate.py), min_k = 2, min_d = 12. 189 survivors.

Evidence trail

Confirmation ladder for this code (dims (2,5,7,4), A = {(0,2,4,3), (1,3,1,3)}, B = {(1,0,3,1), (0,0,2,2)}):

  • screen stages 300 / 3,000 / 20,000 RIS trials: bound flat at 22
  • deep confirm, 100,000 RIS trials: 22; per-side lightest logicals wx = wz = 22
  • adversarial re-check, 10 × 100,000 further RIS samples (1M total): no

    • logical below 22 on either side

  • trusted gate verify/validate_candidate.py: passed — verifies, not refuted

    • (8k RIS), not an exact or WL-equivalent board duplicate

Final claim: d ≤ 22, witness-backed upper bound on both sides (witnesses embedded in codes/560-2-22.json). Not an exact claim.

The weight-6 calibration finding and dead ends are shared with the sweep; see the [[700,2,25]] note for the collapse numbers.

Dead ends

Shared with the sweep: the weight-6 (3+3) arm's screen survivors are unconfirmed pending deep ladders; rank-4 sampling at small moduli needs min_k screening.

Tools

GLM 5.3 Flash (Zed agent), unattended autoresearch pilot (~50 min wall clock, M-series MacBook). Kit: research/kit/search.py, research/kit/surrogate.py (gf2_fast RIS), research/kit/submit.py, verify/validate_candidate.py. The rank-4 constructor is staged on branch quadricycle-constructor @ 0ca843b1; the recipe below is self-contained.

Reproduction

import numpy as np

def build_quad(dims, A, B):
    def shift(r):
        S = np.zeros((r, r), dtype=np.int8)
        i = np.arange(r); S[i, (i + 1) % r] = 1
        return S
    def poly(terms):
        M = np.zeros((int(np.prod(dims)),) * 2, dtype=np.int8)
        for t in terms:
            m = np.array([[1]], dtype=np.int8)
            for r, e in zip(dims, t):
                m = np.kron(m, np.linalg.matrix_power(shift(r), e % r))
            M = (M + m) % 2
        return M
    A_, B_ = poly(A), poly(B)
    return np.hstack([A_, B_]), np.hstack([B_.T, A_.T])

HX, HZ = build_quad((2, 5, 7, 4),
                    [(0, 2, 4, 3), (1, 3, 1, 3)],
                    [(1, 0, 3, 1), (0, 0, 2, 2)])

Sanity anchor: with dims (l, m, 1, 1) this reproduces build_bb from research/kit/bb.py array-exactly on the [[112,2,10]] monomial set.

Parity checks

X-checks 280 (max weight 4) · Z-checks 280 (max weight 4)
H_X (280 checks, sparse supports)
[75, 231, 290, 433] [72, 228, 291, 434] [73, 229, 288, 435] [74, 230, 289, 432] [79, 235, 294, 437] [76, 232, 295, 438] [77, 233, 292, 439] [78, 234, 293, 436] [83, 239, 298, 441] [80, 236, 299, 442] [81, 237, 296, 443] [82, 238, 297, 440] [59, 243, 302, 445] [56, 240, 303, 446] [57, 241, 300, 447] [58, 242, 301, 444] [63, 247, 306, 421] [60, 244, 307, 422] [61, 245, 304, 423] [62, 246, 305, 420] [67, 251, 282, 425] [64, 248, 283, 426] [65, 249, 280, 427] [66, 250, 281, 424] [71, 227, 286, 429] [68, 224, 287, 430] [69, 225, 284, 431] [70, 226, 285, 428] [103, 259, 318, 461] [100, 256, 319, 462] [101, 257, 316, 463] [102, 258, 317, 460] [107, 263, 322, 465] [104, 260, 323, 466] [105, 261, 320, 467] [106, 262, 321, 464] [111, 267, 326, 469] [108, 264, 327, 470] [109, 265, 324, 471] [110, 266, 325, 468] [87, 271, 330, 473] [84, 268, 331, 474] [85, 269, 328, 475] [86, 270, 329, 472] [91, 275, 334, 449] [88, 272, 335, 450] [89, 273, 332, 451] [90, 274, 333, 448] [95, 279, 310, 453] [92, 276, 311, 454] [93, 277, 308, 455] [94, 278, 309, 452] [99, 255, 314, 457] [96, 252, 315, 458] [97, 253, 312, 459] [98, 254, 313, 456] [131, 147, 346, 489] [128, 144, 347, 490] [129, 145, 344, 491] [130, 146, 345, 488] [135, 151, 350, 493] [132, 148, 351, 494] [133, 149, 348, 495] [134, 150, 349, 492] [139, 155, 354, 497] [136, 152, 355, 498] [137, 153, 352, 499] [138, 154, 353, 496] [115, 159, 358, 501] [112, 156, 359, 502] [113, 157, 356, 503] [114, 158, 357, 500] [119, 163, 362, 477] [116, 160, 363, 478] [117, 161, 360, 479] [118, 162, 361, 476] [123, 167, 338, 481] [120, 164, 339, 482] [121, 165, 336, 483] [122, 166, 337, 480] [127, 143, 342, 485] [124, 140, 343, 486] [125, 141, 340, 487] [126, 142, 341, 484] [19, 175, 374, 517] [16, 172, 375, 518] [17, 173, 372, 519] [18, 174, 373, 516] [23, 179, 378, 521] [20, 176, 379, 522] [21, 177, 376, 523] [22, 178, 377, 520] [27, 183, 382, 525] [24, 180, 383, 526] [25, 181, 380, 527] [26, 182, 381, 524] [3, 187, 386, 529] [0, 184, 387, 530] [1, 185, 384, 531] [2, 186, 385, 528] [7, 191, 390, 505] [4, 188, 391, 506] [5, 189, 388, 507] [6, 190, 389, 504] [11, 195, 366, 509] [8, 192, 367, 510] [9, 193, 364, 511] [10, 194, 365, 508] [15, 171, 370, 513] [12, 168, 371, 514] [13, 169, 368, 515] [14, 170, 369, 512] [47, 203, 402, 545] [44, 200, 403, 546] [45, 201, 400, 547] [46, 202, 401, 544] [51, 207, 406, 549] [48, 204, 407, 550] [49, 205, 404, 551] [50, 206, 405, 548] [55, 211, 410, 553] [52, 208, 411, 554] [53, 209, 408, 555] [54, 210, 409, 552] [31, 215, 414, 557] [28, 212, 415, 558] [29, 213, 412, 559] [30, 214, 413, 556] [35, 219, 418, 533] [32, 216, 419, 534] [33, 217, 416, 535] [34, 218, 417, 532] [39, 223, 394, 537] [36, 220, 395, 538] [37, 221, 392, 539] [38, 222, 393, 536] [43, 199, 398, 541] [40, 196, 399, 542] [41, 197, 396, 543] [42, 198, 397, 540] [91, 215, 293, 430] [88, 212, 294, 431] [89, 213, 295, 428] [90, 214, 292, 429] [95, 219, 297, 434] [92, 216, 298, 435] [93, 217, 299, 432] [94, 218, 296, 433] [99, 223, 301, 438] [96, 220, 302, 439] [97, 221, 303, 436] [98, 222, 300, 437] [103, 199, 305, 442] [100, 196, 306, 443] [101, 197, 307, 440] [102, 198, 304, 441] [107, 203, 281, 446] [104, 200, 282, 447] [105, 201, 283, 444] [106, 202, 280, 445] [111, 207, 285, 422] [108, 204, 286, 423] [109, 205, 287, 420] [110, 206, 284, 421] [87, 211, 289, 426] [84, 208, 290, 427] [85, 209, 291, 424] [86, 210, 288, 425] [119, 243, 321, 458] [116, 240, 322, 459] [117, 241, 323, 456] [118, 242, 320, 457] [123, 247, 325, 462] [120, 244, 326, 463] [121, 245, 327, 460] [122, 246, 324, 461] [127, 251, 329, 466] [124, 248, 330, 467] [125, 249, 331, 464] [126, 250, 328, 465] [131, 227, 333, 470] [128, 224, 334, 471] [129, 225, 335, 468] [130, 226, 332, 469] [135, 231, 309, 474] [132, 228, 310, 475] [133, 229, 311, 472] [134, 230, 308, 473] [139, 235, 313, 450] [136, 232, 314, 451] [137, 233, 315, 448] [138, 234, 312, 449] [115, 239, 317, 454] [112, 236, 318, 455] [113, 237, 319, 452] [114, 238, 316, 453] [7, 271, 349, 486] [4, 268, 350, 487] [5, 269, 351, 484] [6, 270, 348, 485] [11, 275, 353, 490] [8, 272, 354, 491] [9, 273, 355, 488] [10, 274, 352, 489] [15, 279, 357, 494] [12, 276, 358, 495] [13, 277, 359, 492] [14, 278, 356, 493] [19, 255, 361, 498] [16, 252, 362, 499] [17, 253, 363, 496] [18, 254, 360, 497] [23, 259, 337, 502] [20, 256, 338, 503] [21, 257, 339, 500] [22, 258, 336, 501] [27, 263, 341, 478] [24, 260, 342, 479] [25, 261, 343, 476] [26, 262, 340, 477] [3, 267, 345, 482] [0, 264, 346, 483] [1, 265, 347, 480] [2, 266, 344, 481] [35, 159, 377, 514] [32, 156, 378, 515] [33, 157, 379, 512] [34, 158, 376, 513] [39, 163, 381, 518] [36, 160, 382, 519] [37, 161, 383, 516] [38, 162, 380, 517] [43, 167, 385, 522] [40, 164, 386, 523] [41, 165, 387, 520] [42, 166, 384, 521] [47, 143, 389, 526] [44, 140, 390, 527] [45, 141, 391, 524] [46, 142, 388, 525] [51, 147, 365, 530] [48, 144, 366, 531] [49, 145, 367, 528] [50, 146, 364, 529] [55, 151, 369, 506] [52, 148, 370, 507] [53, 149, 371, 504] [54, 150, 368, 505] [31, 155, 373, 510] [28, 152, 374, 511] [29, 153, 375, 508] [30, 154, 372, 509] [63, 187, 405, 542] [60, 184, 406, 543] [61, 185, 407, 540] [62, 186, 404, 541] [67, 191, 409, 546] [64, 188, 410, 547] [65, 189, 411, 544] [66, 190, 408, 545] [71, 195, 413, 550] [68, 192, 414, 551] [69, 193, 415, 548] [70, 194, 412, 549] [75, 171, 417, 554] [72, 168, 418, 555] [73, 169, 419, 552] [74, 170, 416, 553] [79, 175, 393, 558] [76, 172, 394, 559] [77, 173, 395, 556] [78, 174, 392, 557] [83, 179, 397, 534] [80, 176, 398, 535] [81, 177, 399, 532] [82, 178, 396, 533] [59, 183, 401, 538] [56, 180, 402, 539] [57, 181, 403, 536] [58, 182, 400, 537]
H_Z (280 checks, sparse supports)
[22, 159, 377, 501] [23, 156, 378, 502] [20, 157, 379, 503] [21, 158, 376, 500] [26, 163, 381, 477] [27, 160, 382, 478] [24, 161, 383, 479] [25, 162, 380, 476] [2, 167, 385, 481] [3, 164, 386, 482] [0, 165, 387, 483] [1, 166, 384, 480] [6, 143, 389, 485] [7, 140, 390, 486] [4, 141, 391, 487] [5, 142, 388, 484] [10, 147, 365, 489] [11, 144, 366, 490] [8, 145, 367, 491] [9, 146, 364, 488] [14, 151, 369, 493] [15, 148, 370, 494] [12, 149, 371, 495] [13, 150, 368, 492] [18, 155, 373, 497] [19, 152, 374, 498] [16, 153, 375, 499] [17, 154, 372, 496] [50, 187, 405, 529] [51, 184, 406, 530] [48, 185, 407, 531] [49, 186, 404, 528] [54, 191, 409, 505] [55, 188, 410, 506] [52, 189, 411, 507] [53, 190, 408, 504] [30, 195, 413, 509] [31, 192, 414, 510] [28, 193, 415, 511] [29, 194, 412, 508] [34, 171, 417, 513] [35, 168, 418, 514] [32, 169, 419, 515] [33, 170, 416, 512] [38, 175, 393, 517] [39, 172, 394, 518] [36, 173, 395, 519] [37, 174, 392, 516] [42, 179, 397, 521] [43, 176, 398, 522] [40, 177, 399, 523] [41, 178, 396, 520] [46, 183, 401, 525] [47, 180, 402, 526] [44, 181, 403, 527] [45, 182, 400, 524] [78, 215, 293, 557] [79, 212, 294, 558] [76, 213, 295, 559] [77, 214, 292, 556] [82, 219, 297, 533] [83, 216, 298, 534] [80, 217, 299, 535] [81, 218, 296, 532] [58, 223, 301, 537] [59, 220, 302, 538] [56, 221, 303, 539] [57, 222, 300, 536] [62, 199, 305, 541] [63, 196, 306, 542] [60, 197, 307, 543] [61, 198, 304, 540] [66, 203, 281, 545] [67, 200, 282, 546] [64, 201, 283, 547] [65, 202, 280, 544] [70, 207, 285, 549] [71, 204, 286, 550] [68, 205, 287, 551] [69, 206, 284, 548] [74, 211, 289, 553] [75, 208, 290, 554] [72, 209, 291, 555] [73, 210, 288, 552] [106, 243, 321, 445] [107, 240, 322, 446] [104, 241, 323, 447] [105, 242, 320, 444] [110, 247, 325, 421] [111, 244, 326, 422] [108, 245, 327, 423] [109, 246, 324, 420] [86, 251, 329, 425] [87, 248, 330, 426] [84, 249, 331, 427] [85, 250, 328, 424] [90, 227, 333, 429] [91, 224, 334, 430] [88, 225, 335, 431] [89, 226, 332, 428] [94, 231, 309, 433] [95, 228, 310, 434] [92, 229, 311, 435] [93, 230, 308, 432] [98, 235, 313, 437] [99, 232, 314, 438] [96, 233, 315, 439] [97, 234, 312, 436] [102, 239, 317, 441] [103, 236, 318, 442] [100, 237, 319, 443] [101, 238, 316, 440] [134, 271, 349, 473] [135, 268, 350, 474] [132, 269, 351, 475] [133, 270, 348, 472] [138, 275, 353, 449] [139, 272, 354, 450] [136, 273, 355, 451] [137, 274, 352, 448] [114, 279, 357, 453] [115, 276, 358, 454] [112, 277, 359, 455] [113, 278, 356, 452] [118, 255, 361, 457] [119, 252, 362, 458] [116, 253, 363, 459] [117, 254, 360, 456] [122, 259, 337, 461] [123, 256, 338, 462] [120, 257, 339, 463] [121, 258, 336, 460] [126, 263, 341, 465] [127, 260, 342, 466] [124, 261, 343, 467] [125, 262, 340, 464] [130, 267, 345, 469] [131, 264, 346, 470] [128, 265, 347, 471] [129, 266, 344, 468] [19, 162, 361, 517] [16, 163, 362, 518] [17, 160, 363, 519] [18, 161, 360, 516] [23, 166, 337, 521] [20, 167, 338, 522] [21, 164, 339, 523] [22, 165, 336, 520] [27, 142, 341, 525] [24, 143, 342, 526] [25, 140, 343, 527] [26, 141, 340, 524] [3, 146, 345, 529] [0, 147, 346, 530] [1, 144, 347, 531] [2, 145, 344, 528] [7, 150, 349, 505] [4, 151, 350, 506] [5, 148, 351, 507] [6, 149, 348, 504] [11, 154, 353, 509] [8, 155, 354, 510] [9, 152, 355, 511] [10, 153, 352, 508] [15, 158, 357, 513] [12, 159, 358, 514] [13, 156, 359, 515] [14, 157, 356, 512] [47, 190, 389, 545] [44, 191, 390, 546] [45, 188, 391, 547] [46, 189, 388, 544] [51, 194, 365, 549] [48, 195, 366, 550] [49, 192, 367, 551] [50, 193, 364, 548] [55, 170, 369, 553] [52, 171, 370, 554] [53, 168, 371, 555] [54, 169, 368, 552] [31, 174, 373, 557] [28, 175, 374, 558] [29, 172, 375, 559] [30, 173, 372, 556] [35, 178, 377, 533] [32, 179, 378, 534] [33, 176, 379, 535] [34, 177, 376, 532] [39, 182, 381, 537] [36, 183, 382, 538] [37, 180, 383, 539] [38, 181, 380, 536] [43, 186, 385, 541] [40, 187, 386, 542] [41, 184, 387, 543] [42, 185, 384, 540] [75, 218, 417, 433] [72, 219, 418, 434] [73, 216, 419, 435] [74, 217, 416, 432] [79, 222, 393, 437] [76, 223, 394, 438] [77, 220, 395, 439] [78, 221, 392, 436] [83, 198, 397, 441] [80, 199, 398, 442] [81, 196, 399, 443] [82, 197, 396, 440] [59, 202, 401, 445] [56, 203, 402, 446] [57, 200, 403, 447] [58, 201, 400, 444] [63, 206, 405, 421] [60, 207, 406, 422] [61, 204, 407, 423] [62, 205, 404, 420] [67, 210, 409, 425] [64, 211, 410, 426] [65, 208, 411, 427] [66, 209, 408, 424] [71, 214, 413, 429] [68, 215, 414, 430] [69, 212, 415, 431] [70, 213, 412, 428] [103, 246, 305, 461] [100, 247, 306, 462] [101, 244, 307, 463] [102, 245, 304, 460] [107, 250, 281, 465] [104, 251, 282, 466] [105, 248, 283, 467] [106, 249, 280, 464] [111, 226, 285, 469] [108, 227, 286, 470] [109, 224, 287, 471] [110, 225, 284, 468] [87, 230, 289, 473] [84, 231, 290, 474] [85, 228, 291, 475] [86, 229, 288, 472] [91, 234, 293, 449] [88, 235, 294, 450] [89, 232, 295, 451] [90, 233, 292, 448] [95, 238, 297, 453] [92, 239, 298, 454] [93, 236, 299, 455] [94, 237, 296, 452] [99, 242, 301, 457] [96, 243, 302, 458] [97, 240, 303, 459] [98, 241, 300, 456] [131, 274, 333, 489] [128, 275, 334, 490] [129, 272, 335, 491] [130, 273, 332, 488] [135, 278, 309, 493] [132, 279, 310, 494] [133, 276, 311, 495] [134, 277, 308, 492] [139, 254, 313, 497] [136, 255, 314, 498] [137, 252, 315, 499] [138, 253, 312, 496] [115, 258, 317, 501] [112, 259, 318, 502] [113, 256, 319, 503] [114, 257, 316, 500] [119, 262, 321, 477] [116, 263, 322, 478] [117, 260, 323, 479] [118, 261, 320, 476] [123, 266, 325, 481] [120, 267, 326, 482] [121, 264, 327, 483] [122, 265, 324, 480] [127, 270, 329, 485] [124, 271, 330, 486] [125, 268, 331, 487] [126, 269, 328, 484]
Code ID 560-2-22 · download JSON · raw on GitHub