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[[576,242,4]] d =
n
576
k
242
d
4
kd²/n
6.722
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 6, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-10
witness operator (support, 4 qubits)
[125, 253, 351, 504]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-10
witness operator (support, 4 qubits)
[20, 23, 523, 524]
certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT
X: no logical < 4 exists; Z: no logical < 4 exists

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 6 · H_Z 8
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×576 (2,2)×2880 (3,2)×14400 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 576 (2,2): 2880 (3,2): 14400 (3,4): 3840
trapping sets H_Z (1,2)×576 (2,2)×4032 (3,2)×28224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 576 (2,2): 4032 (3,2): 28224 (3,4): 8064

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction hyperbolic surface code of arXiv:2610.03682 (L=1 family): finite quotient |G|=576 of the quadrilateral group with corner-rotation orders [3, 4, 3, 4], realized as the transitive generating tuple of involutions in S_8 [0, 1, 2, 3, 5, 4, 7, 6], [0, 1, 2, 4, 3, 5, 7, 6], [3, 4, 6, 0, 1, 7, 2, 5], [4, 5, 6, 7, 0, 1, 2, 3]; a qubit per tile (group element), stabilizers on tiling vertices (X on red, Z on blue vertices of the bipartite tiling), max check weight 8. Enumerated by an exhaustive/sampled search over generating tuples with exact corner-rotation orders (the smooth-surface condition); n, k and the witnesses are recomputed and refutation-tested by the board's verifier.
model MiMo-V2.6-Flash (claimed, not verified)
builds on arXiv:2610.03682
date 2026-10-10
family topological (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

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

[[576,242,4]] hyperbolic surface code, □(3,4,3,4) quotient of order 576

Direction & hypothesis

weight-8 x unrestricted. Nothing at n<=576 in the cell combines k>=242 with d>=4 (nearest: [[452,228,2]] at k=228 d=2, [[376,190,4]] at k=190 d=4).

What was searched

27 finite quotients of the triangle groups △(r,s,t) and the quadrilateral groups □(r,s,t,u) (arXiv:2610.03682's L=1 families), enumerated exhaustively/sample-wise as transitive generating tuples of involutions in S_deg for degrees 6-9 with exact corner-rotation orders (the smooth-surface condition), built into CSS codes, verified against kit css.verify_css, and screened at 400 RIS trials (numpy backend). 12 quotients fell in the n=200-1000 band, 6 were Pareto-non-dominated in their cell against all 1940 CSS board entries and were staged; each survivor then ran a fresh-seed confirmation ladder (500k + 3x1M trials per side, gf2_fast accelerator) before promotion.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 500,000 (seed=101) | 4 | | 1,000,000 (seed=102) | 4 | | 1,000,000 (seed=103) | 4 | | 1,000,000 (seed=104) | 4 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 4 (upper_bound), Z: d <= 4 (upper_bound).

The 400-trial numpy screen's reading was confirmed unchanged at every fresh-seed deep rung; no lighter logical appeared anywhere in the ladder. Distances remain witness-backed upper bounds (d<=), not certified exact.

Dead ends

  • degree-by-degree enumeration at degrees <= 5: no exact-order quotient of any

    • paper tiling exists there; PSL(2,7) (order 168, degree 7) is the only small (4,4,4) quotient and it falls below the n=200 band

  • direct products of known quotients: the product tiling's systole is the

    • minimum over the factors, so low-d building blocks (Bolza, d=2) poison the distance no matter the product's size

  • the paper's tabulated |G|=648 (k=328) and |G|=672 (k=172) instances did not

    • appear within the sweep budgets; full degree-8 exhaustion of △(6,6,6) and △(4,4,4) was still running at promotion time

Tools

Model: MiMo-V2.6-Flash (OpenCode agent session driven by MathysRennela). Repo tooling: research/kit/css, research/kit/surrogate, research/kit/submit, research/kit/coordination, research/kit/escalation, research/kit/promote. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: 8-core Linux; six fresh-seed ladders in parallel, ~4.5M RIS trials per side each on the compiled gf2_fast accelerator.

Reproduction

Permutations of {0..7} written as image tuples (images of 0,1,..., in order): a=[0, 1, 2, 3, 5, 4, 7, 6], b=[0, 1, 2, 4, 3, 5, 7, 6], c=[3, 4, 6, 0, 1, 7, 2, 5], d=[4, 5, 6, 7, 0, 1, 2, 3]. Group them: G = <a, b, c, d> under function composition has order 576, and the corner-rotation orders are exactly (ab, bc, cd, da) = (3, 4, 3, 4), which is also the freeness (smooth-surface) condition. Tiles are the 576 elements of G; a qubit sits on each tile. Corner c of tile g is identified with corner c of tile g∘s for the two sides s adjacent to that corner (corner c lies on sides c and c+1 mod 4), so each vertex's stabilizer support is the set of tiles of its corners. The vertex graph is bipartite; X checks sit on one colour class and Z on the other. n=576; k = n - rank(H_X) - rank(H_Z) over GF(2). The staged entry's exact matrices are the authority; this recipe rebuilds them deterministically.

Parity checks

X-checks 192 (max weight 6) · Z-checks 144 (max weight 8)
H_X (192 checks, sparse supports)
[0, 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, 40, 41] [42, 43, 44, 45, 46, 47] [48, 49, 50, 51, 52, 53] [54, 55, 56, 57, 58, 59] [60, 61, 62, 63, 64, 65] [66, 67, 68, 69, 70, 71] [72, 73, 74, 75, 76, 77] [78, 79, 80, 81, 82, 83] [84, 85, 86, 87, 88, 89] [90, 91, 92, 93, 94, 95] [96, 97, 98, 99, 100, 101] [102, 103, 104, 105, 106, 107] [108, 109, 110, 111, 112, 113] [114, 115, 116, 117, 118, 119] [120, 121, 122, 123, 124, 125] [126, 127, 128, 129, 130, 131] [132, 133, 134, 135, 136, 137] [138, 139, 140, 141, 142, 143] [144, 145, 146, 147, 148, 149] [150, 151, 152, 153, 154, 155] [156, 157, 158, 159, 160, 161] [162, 163, 164, 165, 166, 167] [168, 169, 170, 171, 172, 173] [174, 175, 176, 177, 178, 179] [180, 181, 182, 183, 184, 185] [186, 187, 188, 189, 190, 191] [192, 193, 194, 195, 196, 197] [198, 199, 200, 201, 202, 203] [204, 205, 206, 207, 208, 209] [210, 211, 212, 213, 214, 215] [216, 217, 218, 219, 220, 221] [222, 223, 224, 225, 226, 227] [228, 229, 230, 231, 232, 233] [234, 235, 236, 237, 238, 239] [240, 241, 242, 243, 244, 245] [246, 247, 248, 249, 250, 251] [252, 253, 254, 255, 256, 257] [258, 259, 260, 261, 262, 263] [264, 265, 266, 267, 268, 269] [270, 271, 272, 273, 274, 275] [276, 277, 278, 279, 280, 281] [282, 283, 284, 285, 286, 287] [288, 289, 290, 291, 292, 293] [294, 295, 296, 297, 298, 299] [300, 301, 302, 303, 304, 305] [306, 307, 308, 309, 310, 311] [312, 313, 314, 315, 316, 317] [318, 319, 320, 321, 322, 323] [324, 325, 326, 327, 328, 329] [330, 331, 332, 333, 334, 335] [336, 337, 338, 339, 340, 341] [342, 343, 344, 345, 346, 347] [348, 349, 350, 351, 352, 353] [354, 355, 356, 357, 358, 359] [360, 361, 362, 363, 364, 365] [366, 367, 368, 369, 370, 371] [372, 373, 374, 375, 376, 377] [378, 379, 380, 381, 382, 383] [384, 385, 386, 387, 388, 389] [390, 391, 392, 393, 394, 395] [396, 397, 398, 399, 400, 401] [402, 403, 404, 405, 406, 407] [408, 409, 410, 411, 412, 413] [414, 415, 416, 417, 418, 419] [420, 421, 422, 423, 424, 425] [426, 427, 428, 429, 430, 431] [52, 248, 347, 391, 432, 555] [2, 101, 286, 330, 429, 433] [5, 97, 285, 334, 426, 434] [48, 247, 344, 395, 435, 556] [51, 251, 343, 392, 436, 552] [432, 433, 434, 435, 436, 437] [1, 98, 282, 333, 430, 437] [10, 105, 190, 357, 378, 438] [59, 203, 296, 415, 440, 559] [6, 106, 186, 358, 381, 441] [9, 102, 189, 354, 382, 442] [438, 439, 440, 441, 442, 443] [55, 199, 299, 416, 443, 560] [64, 152, 275, 367, 444, 567] [14, 113, 214, 234, 405, 445] [17, 109, 213, 238, 402, 446] [60, 151, 272, 371, 447, 568] [63, 155, 271, 368, 448, 564] [444, 445, 446, 447, 448, 449] [13, 110, 210, 237, 406, 449] [22, 117, 166, 261, 306, 450] [71, 179, 224, 319, 452, 571] [18, 118, 162, 262, 309, 453] [21, 114, 165, 258, 310, 454] [450, 451, 452, 453, 454, 455] [67, 175, 227, 320, 455, 572] [27, 76, 263, 355, 404, 456] [125, 269, 314, 409, 458, 553] [28, 72, 260, 359, 403, 459] [24, 75, 259, 356, 407, 460] [456, 457, 458, 459, 460, 461] [121, 265, 317, 410, 461, 554] [130, 170, 341, 361, 462, 561] [35, 80, 215, 308, 427, 463] [31, 83, 211, 311, 428, 464] [126, 169, 338, 365, 465, 562] [129, 173, 337, 362, 466, 558] [462, 463, 464, 465, 466, 467] [32, 79, 212, 307, 431, 467] [39, 88, 167, 283, 380, 468] [137, 197, 218, 385, 470, 565] [40, 84, 164, 287, 379, 471] [36, 87, 163, 284, 383, 472] [468, 469, 470, 471, 472, 473] [133, 193, 221, 386, 473, 566] [142, 146, 245, 289, 474, 573] [47, 92, 191, 236, 331, 475] [43, 95, 187, 239, 332, 476] [138, 145, 242, 293, 477, 574] [141, 149, 241, 290, 478, 570] [474, 475, 476, 477, 478, 479] [44, 91, 188, 235, 335, 479] [480, 481, 482, 483, 484, 485] [26, 274, 318, 417, 485, 505] [486, 487, 488, 489, 490, 491] [104, 196, 288, 411, 491, 535] [492, 493, 494, 495, 496, 497] [38, 202, 222, 393, 497, 517] [498, 499, 500, 501, 502, 503] [116, 172, 216, 315, 503, 547] [4, 123, 256, 351, 396, 504] [29, 273, 322, 414, 481, 506] [0, 124, 252, 352, 399, 507] [3, 120, 255, 348, 400, 508] [504, 505, 506, 507, 508, 509] [25, 270, 321, 418, 482, 509] [34, 178, 345, 366, 489, 510] [8, 131, 208, 300, 423, 511] [11, 127, 207, 304, 420, 512] [30, 174, 346, 369, 490, 513] [33, 177, 342, 370, 486, 514] [510, 511, 512, 513, 514, 515] [7, 128, 204, 303, 424, 515] [16, 135, 160, 279, 372, 516] [41, 201, 226, 390, 493, 518] [12, 136, 156, 280, 375, 519] [15, 132, 159, 276, 376, 520] [516, 517, 518, 519, 520, 521] [37, 198, 225, 394, 494, 521] [46, 154, 249, 294, 501, 522] [20, 143, 184, 228, 327, 523] [23, 139, 183, 232, 324, 524] [42, 150, 250, 297, 502, 525] [45, 153, 246, 298, 498, 526] [522, 523, 524, 525, 526, 527] [19, 140, 180, 231, 328, 527] [100, 244, 339, 384, 483, 528] [53, 74, 281, 326, 421, 529] [49, 77, 277, 329, 422, 530] [96, 240, 340, 387, 484, 531] [99, 243, 336, 388, 480, 532] [528, 529, 530, 531, 532, 533] [50, 73, 278, 325, 425, 533] [57, 82, 185, 349, 374, 534] [107, 195, 292, 408, 487, 536] [58, 78, 182, 353, 373, 537] [54, 81, 181, 350, 377, 538] [534, 535, 536, 537, 538, 539] [103, 192, 291, 412, 488, 539] [112, 148, 267, 360, 495, 540] [65, 86, 209, 230, 397, 541] [61, 89, 205, 233, 398, 542] [108, 144, 268, 363, 496, 543] [111, 147, 264, 364, 492, 544] [540, 541, 542, 543, 544, 545] [62, 85, 206, 229, 401, 545] [69, 94, 161, 253, 302, 546] [119, 171, 220, 312, 499, 548] [70, 90, 158, 257, 301, 549] [66, 93, 157, 254, 305, 550] [546, 547, 548, 549, 550, 551] [115, 168, 219, 316, 500, 551] [552, 553, 554, 555, 556, 557] [122, 266, 313, 413, 457, 557] [558, 559, 560, 561, 562, 563] [56, 200, 295, 419, 439, 563] [564, 565, 566, 567, 568, 569] [134, 194, 217, 389, 469, 569] [570, 571, 572, 573, 574, 575] [68, 176, 223, 323, 451, 575]
H_Z (144 checks, sparse supports)
[18, 20, 84, 86, 162, 164, 228, 230] [22, 23, 88, 89, 166, 167, 232, 233] [12, 14, 90, 92, 156, 158, 234, 236] [16, 17, 94, 95, 160, 161, 238, 239] [6, 8, 91, 94, 186, 188, 300, 302] [10, 11, 92, 93, 190, 191, 304, 305] [19, 22, 78, 80, 180, 182, 306, 308] [20, 21, 82, 83, 184, 185, 310, 311] [21, 23, 72, 74, 258, 260, 324, 326] [18, 19, 76, 77, 262, 263, 328, 329] [0, 2, 93, 95, 252, 254, 330, 332] [4, 5, 90, 91, 256, 257, 334, 335] [13, 16, 79, 82, 210, 212, 372, 374] [14, 15, 80, 81, 214, 215, 376, 377] [7, 10, 85, 88, 204, 206, 378, 380] [8, 9, 86, 87, 208, 209, 382, 383] [1, 4, 87, 89, 282, 284, 396, 398] [2, 3, 84, 85, 286, 287, 400, 401] [15, 17, 73, 76, 276, 278, 402, 404] [12, 13, 74, 75, 280, 281, 406, 407] [9, 11, 75, 77, 354, 356, 420, 422] [6, 7, 72, 73, 358, 359, 424, 425] [3, 5, 81, 83, 348, 350, 426, 428] [0, 1, 78, 79, 352, 353, 430, 431] [30, 31, 346, 347, 428, 429, 432, 433] [45, 47, 246, 248, 331, 334, 432, 434] [36, 37, 284, 285, 394, 395, 434, 435] [39, 41, 283, 286, 390, 392, 433, 436] [33, 35, 342, 344, 427, 430, 435, 437] [42, 43, 250, 251, 332, 333, 436, 437] [24, 25, 356, 357, 418, 419, 438, 439] [43, 46, 187, 190, 294, 296, 438, 440] [38, 39, 202, 203, 380, 381, 440, 441] [37, 40, 198, 200, 379, 382, 439, 442] [27, 29, 355, 358, 414, 416, 441, 443] [44, 45, 188, 189, 298, 299, 442, 443] [26, 27, 274, 275, 404, 405, 444, 445] [42, 44, 150, 152, 235, 238, 444, 446] [32, 33, 212, 213, 370, 371, 446, 447] [31, 34, 211, 214, 366, 368, 445, 448] [25, 28, 270, 272, 403, 406, 447, 449] [46, 47, 154, 155, 236, 237, 448, 449] [28, 29, 260, 261, 322, 323, 450, 451] [36, 38, 163, 166, 222, 224, 450, 452] [34, 35, 178, 179, 308, 309, 452, 453] [30, 32, 174, 176, 307, 310, 451, 454] [24, 26, 259, 262, 318, 320, 453, 455] [40, 41, 164, 165, 226, 227, 454, 455] [102, 103, 354, 355, 412, 413, 456, 457] [117, 119, 261, 263, 312, 314, 456, 458] [108, 109, 268, 269, 402, 403, 458, 459] [111, 113, 264, 266, 405, 407, 457, 460] [105, 107, 357, 359, 408, 410, 459, 461] [114, 115, 258, 259, 316, 317, 460, 461] [96, 97, 340, 341, 426, 427, 462, 463] [115, 118, 168, 170, 309, 311, 462, 464] [110, 111, 210, 211, 364, 365, 464, 465] [109, 112, 213, 215, 360, 362, 463, 466] [99, 101, 336, 338, 429, 431, 465, 467] [116, 117, 172, 173, 306, 307, 466, 467] [98, 99, 282, 283, 388, 389, 468, 469] [114, 116, 165, 167, 216, 218, 468, 470] [104, 105, 196, 197, 378, 379, 470, 471] [103, 106, 192, 194, 381, 383, 469, 472] [97, 100, 285, 287, 384, 386, 471, 473] [118, 119, 162, 163, 220, 221, 472, 473] [100, 101, 244, 245, 330, 331, 474, 475] [108, 110, 144, 146, 237, 239, 474, 476] [106, 107, 186, 187, 292, 293, 476, 477] [102, 104, 189, 191, 288, 290, 475, 478] [96, 98, 240, 242, 333, 335, 477, 479] [112, 113, 148, 149, 234, 235, 478, 479] [54, 55, 350, 351, 416, 417, 504, 505] [69, 71, 253, 256, 319, 322, 504, 506] [60, 61, 272, 273, 398, 399, 506, 507] [63, 65, 271, 274, 397, 400, 505, 508] [57, 59, 349, 352, 415, 418, 507, 509] [66, 67, 254, 255, 320, 321, 508, 509] [48, 49, 344, 345, 422, 423, 510, 511] [67, 70, 175, 178, 301, 304, 510, 512] [62, 63, 206, 207, 368, 369, 512, 513] [61, 64, 205, 208, 367, 370, 511, 514] [51, 53, 343, 346, 421, 424, 513, 515] [68, 69, 176, 177, 302, 303, 514, 515] [50, 51, 278, 279, 392, 393, 516, 517] [66, 68, 157, 160, 223, 226, 516, 518] [56, 57, 200, 201, 374, 375, 518, 519] [55, 58, 199, 202, 373, 376, 517, 520] [49, 52, 277, 280, 391, 394, 519, 521] [70, 71, 158, 159, 224, 225, 520, 521] [52, 53, 248, 249, 326, 327, 522, 523] [60, 62, 151, 154, 229, 232, 522, 524] [58, 59, 182, 183, 296, 297, 524, 525] [54, 56, 181, 184, 295, 298, 523, 526] [48, 50, 247, 250, 325, 328, 525, 527] [64, 65, 152, 153, 230, 231, 526, 527] [126, 127, 338, 339, 420, 421, 528, 529] [141, 143, 241, 244, 327, 329, 528, 530] [132, 133, 276, 277, 386, 387, 530, 531] [135, 137, 279, 281, 385, 388, 529, 532] [129, 131, 337, 340, 423, 425, 531, 533] [138, 139, 242, 243, 324, 325, 532, 533] [120, 121, 348, 349, 410, 411, 534, 535] [139, 142, 183, 185, 289, 292, 534, 536] [134, 135, 194, 195, 372, 373, 536, 537] [133, 136, 193, 196, 375, 377, 535, 538] [123, 125, 351, 353, 409, 412, 537, 539] [140, 141, 180, 181, 290, 291, 538, 539] [122, 123, 266, 267, 396, 397, 540, 541] [138, 140, 145, 148, 231, 233, 540, 542] [128, 129, 204, 205, 362, 363, 542, 543] [127, 130, 207, 209, 361, 364, 541, 544] [121, 124, 265, 268, 399, 401, 543, 545] [142, 143, 146, 147, 228, 229, 544, 545] [124, 125, 252, 253, 314, 315, 546, 547] [132, 134, 159, 161, 217, 220, 546, 548] [130, 131, 170, 171, 300, 301, 548, 549] [126, 128, 169, 172, 303, 305, 547, 550] [120, 122, 255, 257, 313, 316, 549, 551] [136, 137, 156, 157, 218, 219, 550, 551] [342, 343, 408, 409, 486, 487, 552, 553] [249, 251, 315, 317, 501, 503, 552, 554] [264, 265, 390, 391, 492, 493, 554, 555] [267, 269, 393, 395, 495, 497, 553, 556] [345, 347, 411, 413, 489, 491, 555, 557] [246, 247, 312, 313, 498, 499, 556, 557] [336, 337, 414, 415, 480, 481, 558, 559] [171, 173, 297, 299, 499, 502, 558, 560] [198, 199, 360, 361, 494, 495, 560, 561] [201, 203, 363, 365, 493, 496, 559, 562] [339, 341, 417, 419, 483, 485, 561, 563] [168, 169, 294, 295, 500, 501, 562, 563] [270, 271, 384, 385, 482, 483, 564, 565] [153, 155, 219, 221, 498, 500, 564, 566] [192, 193, 366, 367, 488, 489, 566, 567] [195, 197, 369, 371, 487, 490, 565, 568] [273, 275, 387, 389, 481, 484, 567, 569] [150, 151, 216, 217, 502, 503, 568, 569] [240, 241, 318, 319, 484, 485, 570, 571] [147, 149, 225, 227, 492, 494, 570, 572] [174, 175, 288, 289, 490, 491, 572, 573] [177, 179, 291, 293, 486, 488, 571, 574] [243, 245, 321, 323, 480, 482, 573, 575] [144, 145, 222, 223, 496, 497, 574, 575]
Code ID 576-242-4 · download JSON · raw on GitHub