[[384,196,4]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 12, w_Z = 12 (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)

[18, 21, 81, 86]

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)

[24, 28, 91, 95]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 12 · H_Z 12

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×384 (2,2)×576 (3,3)×11904 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 384 (2,2): 576 (2,4): 5184 (3,3): 11904 (3,5): 101376 (3,7): 10752

trapping sets H_Z (1,3)×384 (2,2)×576 (3,3)×11904 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 384 (2,2): 576 (2,4): 5184 (3,3): 11904 (3,5): 101376 (3,7): 10752

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction hyperbolic color code of arXiv:2610.03682 (L=1 family): finite quotient |G|=384 of the triangle group with corner-rotation orders [6, 6, 6], realized as the transitive generating tuple of involutions in S_8 [0, 1, 2, 3, 5, 4, 7, 6], [0, 1, 4, 6, 2, 7, 3, 5], [4, 6, 3, 2, 0, 5, 1, 7]; a qubit per tile (group element), stabilizers on tiling vertices (X and Z on every vertex), max check weight 12. 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

[[384,196,4]] hyperbolic color code, △(6,6,6) quotient of order 384

Direction & hypothesis

weight-any x unrestricted. A 51%-rate color code: nothing at n<=384 combines k>=196 with d>=4 in the any-weight cell (nearest: [[376,190,4]] at k=190, w=8; [[372,130,16]] trades down to k=130 for d=16).

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, 4, 6, 2, 7, 3, 5], c=[4, 6, 3, 2, 0, 5, 1, 7]. Group them: G = <a, b, c> under function composition has order 384, and the corner-rotation orders are exactly (ab, bc, ca) = (6, 6, 6), which is also the freeness (smooth-surface) condition. Tiles are the 384 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 3), so each vertex's stabilizer support is the set of tiles of its corners. Both an X and a Z stabilizer are placed on every vertex (supports are identical; pairwise intersections are 0 or 2 tiles, so H_X H_Z^T = 0 automatically). n=384; 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 96 (max weight 12) · Z-checks 96 (max weight 12)

H_X (96 checks, sparse supports)

[0, 2, 13, 15, 17, 22, 25, 30, 34, 37, 42, 45] [0, 17, 61, 82, 100, 129, 153, 162, 195, 216, 296, 307] [0, 2, 8, 10, 208, 212, 216, 220, 256, 260, 264, 268] [1, 6, 11, 12, 16, 20, 26, 29, 35, 39, 41, 46] [1, 16, 60, 83, 101, 128, 152, 163, 194, 217, 297, 306] [1, 6, 9, 14, 209, 214, 217, 222, 352, 356, 360, 364] [2, 25, 63, 90, 102, 137, 155, 170, 243, 264, 344, 355] [3, 4, 9, 14, 18, 21, 24, 28, 33, 38, 43, 47] [3, 24, 62, 91, 103, 136, 154, 171, 242, 265, 345, 354] [3, 4, 11, 12, 257, 262, 265, 270, 304, 308, 312, 316] [4, 33, 57, 66, 96, 113, 157, 178, 200, 211, 291, 312] [5, 7, 8, 10, 19, 23, 27, 31, 32, 36, 40, 44] [5, 32, 56, 67, 97, 112, 156, 179, 201, 210, 290, 313] [5, 7, 13, 15, 305, 310, 313, 318, 353, 358, 361, 366] [6, 41, 59, 74, 98, 121, 159, 186, 248, 259, 339, 360] [7, 40, 58, 75, 99, 120, 158, 187, 249, 258, 338, 361] [8, 19, 53, 80, 105, 114, 148, 177, 193, 208, 298, 315] [9, 18, 52, 81, 104, 115, 149, 176, 192, 209, 299, 314] [10, 27, 55, 88, 107, 122, 150, 185, 241, 256, 346, 363] [11, 26, 54, 89, 106, 123, 151, 184, 240, 257, 347, 362] [12, 35, 49, 64, 109, 130, 144, 161, 202, 219, 289, 304] [13, 34, 48, 65, 108, 131, 145, 160, 203, 218, 288, 305] [14, 43, 51, 72, 111, 138, 146, 169, 250, 267, 337, 352] [15, 42, 50, 73, 110, 139, 147, 168, 251, 266, 336, 353] [16, 20, 32, 36, 112, 116, 128, 132, 272, 276, 280, 284] [17, 22, 33, 38, 113, 118, 129, 134, 368, 372, 376, 380] [18, 21, 34, 37, 160, 164, 176, 180, 273, 278, 281, 286] [19, 23, 35, 39, 161, 166, 177, 182, 369, 374, 377, 382] [20, 29, 87, 94, 198, 233, 247, 280, 301, 322, 348, 371] [21, 28, 86, 95, 199, 232, 246, 281, 300, 323, 349, 370] [22, 45, 78, 85, 196, 225, 252, 275, 303, 330, 343, 376] [23, 44, 79, 84, 197, 224, 253, 274, 302, 331, 342, 377] [24, 28, 40, 44, 120, 124, 136, 140, 224, 228, 232, 236] [25, 30, 41, 46, 121, 126, 137, 142, 320, 324, 328, 332] [26, 29, 42, 45, 168, 172, 184, 188, 225, 230, 233, 238] [27, 31, 43, 47, 169, 174, 185, 190, 321, 326, 329, 334] [30, 37, 70, 93, 204, 227, 244, 273, 295, 328, 351, 378] [31, 36, 71, 92, 205, 226, 245, 272, 294, 329, 350, 379] [38, 47, 69, 76, 207, 234, 254, 283, 292, 321, 341, 368] [39, 46, 68, 77, 206, 235, 255, 282, 293, 320, 340, 369] [48, 50, 61, 63, 65, 70, 73, 78, 82, 85, 90, 93] [48, 50, 56, 58, 210, 213, 218, 221, 258, 261, 266, 269] [49, 54, 59, 60, 64, 68, 74, 77, 83, 87, 89, 94] [49, 54, 57, 62, 211, 215, 219, 223, 354, 357, 362, 365] [51, 52, 57, 62, 66, 69, 72, 76, 81, 86, 91, 95] [51, 52, 59, 60, 259, 263, 267, 271, 306, 309, 314, 317] [53, 55, 56, 58, 67, 71, 75, 79, 80, 84, 88, 92] [53, 55, 61, 63, 307, 311, 315, 319, 355, 359, 363, 367] [64, 68, 80, 84, 114, 117, 130, 133, 274, 277, 282, 285] [65, 70, 81, 86, 115, 119, 131, 135, 370, 373, 378, 381] [66, 69, 82, 85, 162, 165, 178, 181, 275, 279, 283, 287] [67, 71, 83, 87, 163, 167, 179, 183, 371, 375, 379, 383] [72, 76, 88, 92, 122, 125, 138, 141, 226, 229, 234, 237] [73, 78, 89, 94, 123, 127, 139, 143, 322, 325, 330, 333] [74, 77, 90, 93, 170, 173, 186, 189, 227, 231, 235, 239] [75, 79, 91, 95, 171, 175, 187, 191, 323, 327, 331, 335] [96, 98, 109, 111, 113, 118, 121, 126, 130, 133, 138, 141] [96, 98, 104, 106, 192, 196, 200, 204, 240, 244, 248, 252] [97, 102, 107, 108, 112, 116, 122, 125, 131, 135, 137, 142] [97, 102, 105, 110, 193, 198, 201, 206, 336, 340, 344, 348] [99, 100, 105, 110, 114, 117, 120, 124, 129, 134, 139, 143] [99, 100, 107, 108, 241, 246, 249, 254, 288, 292, 296, 300] [101, 103, 104, 106, 115, 119, 123, 127, 128, 132, 136, 140] [101, 103, 109, 111, 289, 294, 297, 302, 337, 342, 345, 350] [116, 125, 183, 190, 214, 237, 263, 284, 317, 326, 364, 375] [117, 124, 182, 191, 215, 236, 262, 285, 316, 327, 365, 374] [118, 141, 174, 181, 212, 229, 268, 279, 319, 334, 359, 380] [119, 140, 175, 180, 213, 228, 269, 278, 318, 335, 358, 381] [126, 133, 166, 189, 220, 231, 260, 277, 311, 332, 367, 382] [127, 132, 167, 188, 221, 230, 261, 276, 310, 333, 366, 383] [134, 143, 165, 172, 223, 238, 270, 287, 308, 325, 357, 372] [135, 142, 164, 173, 222, 239, 271, 286, 309, 324, 356, 373] [144, 146, 157, 159, 161, 166, 169, 174, 178, 181, 186, 189] [144, 146, 152, 154, 194, 197, 202, 205, 242, 245, 250, 253] [145, 150, 155, 156, 160, 164, 170, 173, 179, 183, 185, 190] [145, 150, 153, 158, 195, 199, 203, 207, 338, 341, 346, 349] [147, 148, 153, 158, 162, 165, 168, 172, 177, 182, 187, 191] [147, 148, 155, 156, 243, 247, 251, 255, 290, 293, 298, 301] [149, 151, 152, 154, 163, 167, 171, 175, 176, 180, 184, 188] [149, 151, 157, 159, 291, 295, 299, 303, 339, 343, 347, 351] [192, 196, 203, 207, 209, 214, 218, 221, 225, 230, 234, 237] [193, 198, 202, 205, 208, 212, 219, 223, 226, 229, 233, 238] [194, 197, 201, 206, 210, 213, 217, 222, 224, 228, 235, 239] [195, 199, 200, 204, 211, 215, 216, 220, 227, 231, 232, 236] [240, 244, 251, 255, 257, 262, 266, 269, 273, 278, 282, 285] [241, 246, 250, 253, 256, 260, 267, 271, 274, 277, 281, 286] [242, 245, 249, 254, 258, 261, 265, 270, 272, 276, 283, 287] [243, 247, 248, 252, 259, 263, 264, 268, 275, 279, 280, 284] [288, 292, 299, 303, 305, 310, 314, 317, 321, 326, 330, 333] [289, 294, 298, 301, 304, 308, 315, 319, 322, 325, 329, 334] [290, 293, 297, 302, 306, 309, 313, 318, 320, 324, 331, 335] [291, 295, 296, 300, 307, 311, 312, 316, 323, 327, 328, 332] [336, 340, 347, 351, 353, 358, 362, 365, 369, 374, 378, 381] [337, 342, 346, 349, 352, 356, 363, 367, 370, 373, 377, 382] [338, 341, 345, 350, 354, 357, 361, 366, 368, 372, 379, 383] [339, 343, 344, 348, 355, 359, 360, 364, 371, 375, 376, 380]

H_Z (96 checks, sparse supports)

[0, 2, 13, 15, 17, 22, 25, 30, 34, 37, 42, 45] [0, 17, 61, 82, 100, 129, 153, 162, 195, 216, 296, 307] [0, 2, 8, 10, 208, 212, 216, 220, 256, 260, 264, 268] [1, 6, 11, 12, 16, 20, 26, 29, 35, 39, 41, 46] [1, 16, 60, 83, 101, 128, 152, 163, 194, 217, 297, 306] [1, 6, 9, 14, 209, 214, 217, 222, 352, 356, 360, 364] [2, 25, 63, 90, 102, 137, 155, 170, 243, 264, 344, 355] [3, 4, 9, 14, 18, 21, 24, 28, 33, 38, 43, 47] [3, 24, 62, 91, 103, 136, 154, 171, 242, 265, 345, 354] [3, 4, 11, 12, 257, 262, 265, 270, 304, 308, 312, 316] [4, 33, 57, 66, 96, 113, 157, 178, 200, 211, 291, 312] [5, 7, 8, 10, 19, 23, 27, 31, 32, 36, 40, 44] [5, 32, 56, 67, 97, 112, 156, 179, 201, 210, 290, 313] [5, 7, 13, 15, 305, 310, 313, 318, 353, 358, 361, 366] [6, 41, 59, 74, 98, 121, 159, 186, 248, 259, 339, 360] [7, 40, 58, 75, 99, 120, 158, 187, 249, 258, 338, 361] [8, 19, 53, 80, 105, 114, 148, 177, 193, 208, 298, 315] [9, 18, 52, 81, 104, 115, 149, 176, 192, 209, 299, 314] [10, 27, 55, 88, 107, 122, 150, 185, 241, 256, 346, 363] [11, 26, 54, 89, 106, 123, 151, 184, 240, 257, 347, 362] [12, 35, 49, 64, 109, 130, 144, 161, 202, 219, 289, 304] [13, 34, 48, 65, 108, 131, 145, 160, 203, 218, 288, 305] [14, 43, 51, 72, 111, 138, 146, 169, 250, 267, 337, 352] [15, 42, 50, 73, 110, 139, 147, 168, 251, 266, 336, 353] [16, 20, 32, 36, 112, 116, 128, 132, 272, 276, 280, 284] [17, 22, 33, 38, 113, 118, 129, 134, 368, 372, 376, 380] [18, 21, 34, 37, 160, 164, 176, 180, 273, 278, 281, 286] [19, 23, 35, 39, 161, 166, 177, 182, 369, 374, 377, 382] [20, 29, 87, 94, 198, 233, 247, 280, 301, 322, 348, 371] [21, 28, 86, 95, 199, 232, 246, 281, 300, 323, 349, 370] [22, 45, 78, 85, 196, 225, 252, 275, 303, 330, 343, 376] [23, 44, 79, 84, 197, 224, 253, 274, 302, 331, 342, 377] [24, 28, 40, 44, 120, 124, 136, 140, 224, 228, 232, 236] [25, 30, 41, 46, 121, 126, 137, 142, 320, 324, 328, 332] [26, 29, 42, 45, 168, 172, 184, 188, 225, 230, 233, 238] [27, 31, 43, 47, 169, 174, 185, 190, 321, 326, 329, 334] [30, 37, 70, 93, 204, 227, 244, 273, 295, 328, 351, 378] [31, 36, 71, 92, 205, 226, 245, 272, 294, 329, 350, 379] [38, 47, 69, 76, 207, 234, 254, 283, 292, 321, 341, 368] [39, 46, 68, 77, 206, 235, 255, 282, 293, 320, 340, 369] [48, 50, 61, 63, 65, 70, 73, 78, 82, 85, 90, 93] [48, 50, 56, 58, 210, 213, 218, 221, 258, 261, 266, 269] [49, 54, 59, 60, 64, 68, 74, 77, 83, 87, 89, 94] [49, 54, 57, 62, 211, 215, 219, 223, 354, 357, 362, 365] [51, 52, 57, 62, 66, 69, 72, 76, 81, 86, 91, 95] [51, 52, 59, 60, 259, 263, 267, 271, 306, 309, 314, 317] [53, 55, 56, 58, 67, 71, 75, 79, 80, 84, 88, 92] [53, 55, 61, 63, 307, 311, 315, 319, 355, 359, 363, 367] [64, 68, 80, 84, 114, 117, 130, 133, 274, 277, 282, 285] [65, 70, 81, 86, 115, 119, 131, 135, 370, 373, 378, 381] [66, 69, 82, 85, 162, 165, 178, 181, 275, 279, 283, 287] [67, 71, 83, 87, 163, 167, 179, 183, 371, 375, 379, 383] [72, 76, 88, 92, 122, 125, 138, 141, 226, 229, 234, 237] [73, 78, 89, 94, 123, 127, 139, 143, 322, 325, 330, 333] [74, 77, 90, 93, 170, 173, 186, 189, 227, 231, 235, 239] [75, 79, 91, 95, 171, 175, 187, 191, 323, 327, 331, 335] [96, 98, 109, 111, 113, 118, 121, 126, 130, 133, 138, 141] [96, 98, 104, 106, 192, 196, 200, 204, 240, 244, 248, 252] [97, 102, 107, 108, 112, 116, 122, 125, 131, 135, 137, 142] [97, 102, 105, 110, 193, 198, 201, 206, 336, 340, 344, 348] [99, 100, 105, 110, 114, 117, 120, 124, 129, 134, 139, 143] [99, 100, 107, 108, 241, 246, 249, 254, 288, 292, 296, 300] [101, 103, 104, 106, 115, 119, 123, 127, 128, 132, 136, 140] [101, 103, 109, 111, 289, 294, 297, 302, 337, 342, 345, 350] [116, 125, 183, 190, 214, 237, 263, 284, 317, 326, 364, 375] [117, 124, 182, 191, 215, 236, 262, 285, 316, 327, 365, 374] [118, 141, 174, 181, 212, 229, 268, 279, 319, 334, 359, 380] [119, 140, 175, 180, 213, 228, 269, 278, 318, 335, 358, 381] [126, 133, 166, 189, 220, 231, 260, 277, 311, 332, 367, 382] [127, 132, 167, 188, 221, 230, 261, 276, 310, 333, 366, 383] [134, 143, 165, 172, 223, 238, 270, 287, 308, 325, 357, 372] [135, 142, 164, 173, 222, 239, 271, 286, 309, 324, 356, 373] [144, 146, 157, 159, 161, 166, 169, 174, 178, 181, 186, 189] [144, 146, 152, 154, 194, 197, 202, 205, 242, 245, 250, 253] [145, 150, 155, 156, 160, 164, 170, 173, 179, 183, 185, 190] [145, 150, 153, 158, 195, 199, 203, 207, 338, 341, 346, 349] [147, 148, 153, 158, 162, 165, 168, 172, 177, 182, 187, 191] [147, 148, 155, 156, 243, 247, 251, 255, 290, 293, 298, 301] [149, 151, 152, 154, 163, 167, 171, 175, 176, 180, 184, 188] [149, 151, 157, 159, 291, 295, 299, 303, 339, 343, 347, 351] [192, 196, 203, 207, 209, 214, 218, 221, 225, 230, 234, 237] [193, 198, 202, 205, 208, 212, 219, 223, 226, 229, 233, 238] [194, 197, 201, 206, 210, 213, 217, 222, 224, 228, 235, 239] [195, 199, 200, 204, 211, 215, 216, 220, 227, 231, 232, 236] [240, 244, 251, 255, 257, 262, 266, 269, 273, 278, 282, 285] [241, 246, 250, 253, 256, 260, 267, 271, 274, 277, 281, 286] [242, 245, 249, 254, 258, 261, 265, 270, 272, 276, 283, 287] [243, 247, 248, 252, 259, 263, 264, 268, 275, 279, 280, 284] [288, 292, 299, 303, 305, 310, 314, 317, 321, 326, 330, 333] [289, 294, 298, 301, 304, 308, 315, 319, 322, 325, 329, 334] [290, 293, 297, 302, 306, 309, 313, 318, 320, 324, 331, 335] [291, 295, 296, 300, 307, 311, 312, 316, 323, 327, 328, 332] [336, 340, 347, 351, 353, 358, 362, 365, 369, 374, 378, 381] [337, 342, 346, 349, 352, 356, 363, 367, 370, 373, 377, 382] [338, 341, 345, 350, 354, 357, 361, 366, 368, 372, 379, 383] [339, 343, 344, 348, 355, 359, 360, 364, 371, 375, 376, 380]

Code ID 384-196-4 · download JSON · raw on GitHub