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[[308,10,26]] d ≤
n
308
k
10
d
26
kd²/n
21.948
w
8
X/Z
1.04

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Distance

X/Z asymmetry 1.04 · d_X ≤ 27, d_Z ≤ 26 · w_X = 8, 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 27 · witness weight 27 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 3×108 trials · survived 3×108 trials · 2026-09-27
witness operator (support, 27 qubits)
[1, 2, 11, 12, 29, 30, 38, 39, 46, 58, 75, 76, 83, 84, 95, 101, 110, 128, 138, 139, 148, 161, 197, 219, 249, 269, 290]
d_Z 26 · witness weight 26 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 3×108 trials · survived 3×108 trials · 2026-09-27
witness operator (support, 26 qubits)
[8, 9, 12, 15, 51, 64, 76, 78, 81, 92, 101, 103, 117, 122, 127, 137, 147, 159, 180, 184, 220, 260, 270, 276, 285, 301]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×308 (2,6)×4312 (3,6)×2464 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 308 (2,6): 4312 (3,6): 2464 (3,8): 83160 (3,10): 8624
trapping sets H_Z (1,4)×308 (2,6)×4312 (3,6)×2464 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 308 (2,6): 4312 (3,6): 2464 (3,8): 83160 (3,10): 8624

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Periodic bivariate-bicycle code on Z_14 x Z_11 (n = 2*l*m = 308): x = S_14 tensor I_11, y = I_14 tensor S_11 (cyclic shifts), qubit index i*m + j in each block; A = x0y0 + x2y3 + x5y6 + x13y6, B = x0y0 + x5y9 + x10y9 + x11y7; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(14, 11, [[0, 0], [2, 3], [5, 6], [13, 6]], [[0, 0], [5, 9], [10, 9], [11, 7]])). gcd(14, 11) = 1, so Z_14 x Z_11 is cyclic of order 154 and the code is the cyclic generalized-bicycle code over Z_154 with a(z) = z0 + z58 + z61 + z83, b(z) = z0 + z75 + z95 + z108 (CRT relabeling x^a y^b -> z^t, t = a mod 14, t = b mod 11).
model Claude Claude Fable 5.1 (Claude Code) (claimed, not verified)
date 2026-09-27
family other (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

[[308,10,26]] weight-8 coprime bivariate-bicycle code

Construction

Periodic bivariate-bicycle code on Z_14 x Z_11 (n = 2*l*m = 308): x = S_14 tensor I_11, y = I_14 tensor S_11 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^2y^3 + x^5y^6 + x^13y^6, B = x^0y^0 + x^5y^9 + x^10y^9 + x^11y^7; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(14, 11, [[0, 0], [2, 3], [5, 6], [13, 6]], [[0, 0], [5, 9], [10, 9], [11, 7]])). gcd(14, 11) = 1, so Z_14 x Z_11 is cyclic of order 154 and the code is the cyclic generalized-bicycle code over Z_154 with a(z) = z^0 + z^58 + z^61 + z^83, b(z) = z^0 + z^75 + z^95 + z^108 (CRT relabeling x^a y^b -> z^t, t = a mod 14, t = b mod 11).

Every check has weight exactly 8.

Distance evidence

The distance is a witness-backed upper bound. Every witness in the submission was validated against the published check matrices before filing.

| trials per side | lightest logical found | |---|---| | 300 | 36 | | 2,000 | 28 | | 20,000 | 26 | | 300,000,000 | X 27, Z 26 |

The deeper passes did not lower the claim.

What is not claimed

This code carries no circuit-level or logical-error-rate measurement. It came from a search whose objective was a memory-experiment comparison against the rotated surface code, but it did not reach that stage, so nothing here speaks to its performance under a decoder. The submission is a claim about n, k, d, and check weight only.

The distance is an upper bound, not a certificate. A lighter logical operator would refute it, and the witnesses are published so that anyone can try.

Reproduction

import sys; sys.path.insert(0, "research/kit")
from bb import build_bb
HX, HZ = build_bb(l=14, m=11, A_terms=[[0, 0], [2, 3], [5, 6], [13, 6]], B_terms=[[0, 0], [5, 9], [10, 9], [11, 7]])

Parity checks

X-checks 154 (max weight 8) · Z-checks 154 (max weight 8)
H_X (154 checks, sparse supports)
[0, 25, 61, 149, 154, 218, 273, 282] [1, 26, 62, 150, 155, 219, 274, 283] [2, 27, 63, 151, 156, 209, 264, 284] [3, 28, 64, 152, 157, 210, 265, 285] [4, 29, 65, 153, 158, 211, 266, 275] [5, 30, 55, 143, 159, 212, 267, 276] [6, 31, 56, 144, 160, 213, 268, 277] [7, 32, 57, 145, 161, 214, 269, 278] [8, 22, 58, 146, 162, 215, 270, 279] [9, 23, 59, 147, 163, 216, 271, 280] [10, 24, 60, 148, 164, 217, 272, 281] [6, 11, 36, 72, 165, 229, 284, 293] [7, 12, 37, 73, 166, 230, 285, 294] [8, 13, 38, 74, 167, 220, 275, 295] [9, 14, 39, 75, 168, 221, 276, 296] [10, 15, 40, 76, 169, 222, 277, 286] [0, 16, 41, 66, 170, 223, 278, 287] [1, 17, 42, 67, 171, 224, 279, 288] [2, 18, 43, 68, 172, 225, 280, 289] [3, 19, 33, 69, 173, 226, 281, 290] [4, 20, 34, 70, 174, 227, 282, 291] [5, 21, 35, 71, 175, 228, 283, 292] [17, 22, 47, 83, 176, 240, 295, 304] [18, 23, 48, 84, 177, 241, 296, 305] [19, 24, 49, 85, 178, 231, 286, 306] [20, 25, 50, 86, 179, 232, 287, 307] [21, 26, 51, 87, 180, 233, 288, 297] [11, 27, 52, 77, 181, 234, 289, 298] [12, 28, 53, 78, 182, 235, 290, 299] [13, 29, 54, 79, 183, 236, 291, 300] [14, 30, 44, 80, 184, 237, 292, 301] [15, 31, 45, 81, 185, 238, 293, 302] [16, 32, 46, 82, 186, 239, 294, 303] [28, 33, 58, 94, 161, 187, 251, 306] [29, 34, 59, 95, 162, 188, 252, 307] [30, 35, 60, 96, 163, 189, 242, 297] [31, 36, 61, 97, 164, 190, 243, 298] [32, 37, 62, 98, 154, 191, 244, 299] [22, 38, 63, 88, 155, 192, 245, 300] [23, 39, 64, 89, 156, 193, 246, 301] [24, 40, 65, 90, 157, 194, 247, 302] [25, 41, 55, 91, 158, 195, 248, 303] [26, 42, 56, 92, 159, 196, 249, 304] [27, 43, 57, 93, 160, 197, 250, 305] [39, 44, 69, 105, 163, 172, 198, 262] [40, 45, 70, 106, 164, 173, 199, 263] [41, 46, 71, 107, 154, 174, 200, 253] [42, 47, 72, 108, 155, 175, 201, 254] [43, 48, 73, 109, 156, 165, 202, 255] [33, 49, 74, 99, 157, 166, 203, 256] [34, 50, 75, 100, 158, 167, 204, 257] [35, 51, 76, 101, 159, 168, 205, 258] [36, 52, 66, 102, 160, 169, 206, 259] [37, 53, 67, 103, 161, 170, 207, 260] [38, 54, 68, 104, 162, 171, 208, 261] [50, 55, 80, 116, 174, 183, 209, 273] [51, 56, 81, 117, 175, 184, 210, 274] [52, 57, 82, 118, 165, 185, 211, 264] [53, 58, 83, 119, 166, 186, 212, 265] [54, 59, 84, 120, 167, 176, 213, 266] [44, 60, 85, 110, 168, 177, 214, 267] [45, 61, 86, 111, 169, 178, 215, 268] [46, 62, 87, 112, 170, 179, 216, 269] [47, 63, 77, 113, 171, 180, 217, 270] [48, 64, 78, 114, 172, 181, 218, 271] [49, 65, 79, 115, 173, 182, 219, 272] [61, 66, 91, 127, 185, 194, 220, 284] [62, 67, 92, 128, 186, 195, 221, 285] [63, 68, 93, 129, 176, 196, 222, 275] [64, 69, 94, 130, 177, 197, 223, 276] [65, 70, 95, 131, 178, 187, 224, 277] [55, 71, 96, 121, 179, 188, 225, 278] [56, 72, 97, 122, 180, 189, 226, 279] [57, 73, 98, 123, 181, 190, 227, 280] [58, 74, 88, 124, 182, 191, 228, 281] [59, 75, 89, 125, 183, 192, 229, 282] [60, 76, 90, 126, 184, 193, 230, 283] [72, 77, 102, 138, 196, 205, 231, 295] [73, 78, 103, 139, 197, 206, 232, 296] [74, 79, 104, 140, 187, 207, 233, 286] [75, 80, 105, 141, 188, 208, 234, 287] [76, 81, 106, 142, 189, 198, 235, 288] [66, 82, 107, 132, 190, 199, 236, 289] [67, 83, 108, 133, 191, 200, 237, 290] [68, 84, 109, 134, 192, 201, 238, 291] [69, 85, 99, 135, 193, 202, 239, 292] [70, 86, 100, 136, 194, 203, 240, 293] [71, 87, 101, 137, 195, 204, 241, 294] [83, 88, 113, 149, 207, 216, 242, 306] [84, 89, 114, 150, 208, 217, 243, 307] [85, 90, 115, 151, 198, 218, 244, 297] [86, 91, 116, 152, 199, 219, 245, 298] [87, 92, 117, 153, 200, 209, 246, 299] [77, 93, 118, 143, 201, 210, 247, 300] [78, 94, 119, 144, 202, 211, 248, 301] [79, 95, 120, 145, 203, 212, 249, 302] [80, 96, 110, 146, 204, 213, 250, 303] [81, 97, 111, 147, 205, 214, 251, 304] [82, 98, 112, 148, 206, 215, 252, 305] [6, 94, 99, 124, 163, 218, 227, 253] [7, 95, 100, 125, 164, 219, 228, 254] [8, 96, 101, 126, 154, 209, 229, 255] [9, 97, 102, 127, 155, 210, 230, 256] [10, 98, 103, 128, 156, 211, 220, 257] [0, 88, 104, 129, 157, 212, 221, 258] [1, 89, 105, 130, 158, 213, 222, 259] [2, 90, 106, 131, 159, 214, 223, 260] [3, 91, 107, 121, 160, 215, 224, 261] [4, 92, 108, 122, 161, 216, 225, 262] [5, 93, 109, 123, 162, 217, 226, 263] [17, 105, 110, 135, 174, 229, 238, 264] [18, 106, 111, 136, 175, 230, 239, 265] [19, 107, 112, 137, 165, 220, 240, 266] [20, 108, 113, 138, 166, 221, 241, 267] [21, 109, 114, 139, 167, 222, 231, 268] [11, 99, 115, 140, 168, 223, 232, 269] [12, 100, 116, 141, 169, 224, 233, 270] [13, 101, 117, 142, 170, 225, 234, 271] [14, 102, 118, 132, 171, 226, 235, 272] [15, 103, 119, 133, 172, 227, 236, 273] [16, 104, 120, 134, 173, 228, 237, 274] [28, 116, 121, 146, 185, 240, 249, 275] [29, 117, 122, 147, 186, 241, 250, 276] [30, 118, 123, 148, 176, 231, 251, 277] [31, 119, 124, 149, 177, 232, 252, 278] [32, 120, 125, 150, 178, 233, 242, 279] [22, 110, 126, 151, 179, 234, 243, 280] [23, 111, 127, 152, 180, 235, 244, 281] [24, 112, 128, 153, 181, 236, 245, 282] [25, 113, 129, 143, 182, 237, 246, 283] [26, 114, 130, 144, 183, 238, 247, 284] [27, 115, 131, 145, 184, 239, 248, 285] [3, 39, 127, 132, 196, 251, 260, 286] [4, 40, 128, 133, 197, 252, 261, 287] [5, 41, 129, 134, 187, 242, 262, 288] [6, 42, 130, 135, 188, 243, 263, 289] [7, 43, 131, 136, 189, 244, 253, 290] [8, 33, 121, 137, 190, 245, 254, 291] [9, 34, 122, 138, 191, 246, 255, 292] [10, 35, 123, 139, 192, 247, 256, 293] [0, 36, 124, 140, 193, 248, 257, 294] [1, 37, 125, 141, 194, 249, 258, 295] [2, 38, 126, 142, 195, 250, 259, 296] [14, 50, 138, 143, 207, 262, 271, 297] [15, 51, 139, 144, 208, 263, 272, 298] [16, 52, 140, 145, 198, 253, 273, 299] [17, 53, 141, 146, 199, 254, 274, 300] [18, 54, 142, 147, 200, 255, 264, 301] [19, 44, 132, 148, 201, 256, 265, 302] [20, 45, 133, 149, 202, 257, 266, 303] [21, 46, 134, 150, 203, 258, 267, 304] [11, 47, 135, 151, 204, 259, 268, 305] [12, 48, 136, 152, 205, 260, 269, 306] [13, 49, 137, 153, 206, 261, 270, 307]
H_Z (154 checks, sparse supports)
[0, 37, 46, 101, 154, 170, 258, 294] [1, 38, 47, 102, 155, 171, 259, 295] [2, 39, 48, 103, 156, 172, 260, 296] [3, 40, 49, 104, 157, 173, 261, 286] [4, 41, 50, 105, 158, 174, 262, 287] [5, 42, 51, 106, 159, 175, 263, 288] [6, 43, 52, 107, 160, 165, 253, 289] [7, 33, 53, 108, 161, 166, 254, 290] [8, 34, 54, 109, 162, 167, 255, 291] [9, 35, 44, 99, 163, 168, 256, 292] [10, 36, 45, 100, 164, 169, 257, 293] [11, 48, 57, 112, 165, 181, 269, 305] [12, 49, 58, 113, 166, 182, 270, 306] [13, 50, 59, 114, 167, 183, 271, 307] [14, 51, 60, 115, 168, 184, 272, 297] [15, 52, 61, 116, 169, 185, 273, 298] [16, 53, 62, 117, 170, 186, 274, 299] [17, 54, 63, 118, 171, 176, 264, 300] [18, 44, 64, 119, 172, 177, 265, 301] [19, 45, 65, 120, 173, 178, 266, 302] [20, 46, 55, 110, 174, 179, 267, 303] [21, 47, 56, 111, 175, 180, 268, 304] [22, 59, 68, 123, 162, 176, 192, 280] [23, 60, 69, 124, 163, 177, 193, 281] [24, 61, 70, 125, 164, 178, 194, 282] [25, 62, 71, 126, 154, 179, 195, 283] [26, 63, 72, 127, 155, 180, 196, 284] [27, 64, 73, 128, 156, 181, 197, 285] [28, 65, 74, 129, 157, 182, 187, 275] [29, 55, 75, 130, 158, 183, 188, 276] [30, 56, 76, 131, 159, 184, 189, 277] [31, 57, 66, 121, 160, 185, 190, 278] [32, 58, 67, 122, 161, 186, 191, 279] [33, 70, 79, 134, 173, 187, 203, 291] [34, 71, 80, 135, 174, 188, 204, 292] [35, 72, 81, 136, 175, 189, 205, 293] [36, 73, 82, 137, 165, 190, 206, 294] [37, 74, 83, 138, 166, 191, 207, 295] [38, 75, 84, 139, 167, 192, 208, 296] [39, 76, 85, 140, 168, 193, 198, 286] [40, 66, 86, 141, 169, 194, 199, 287] [41, 67, 87, 142, 170, 195, 200, 288] [42, 68, 77, 132, 171, 196, 201, 289] [43, 69, 78, 133, 172, 197, 202, 290] [44, 81, 90, 145, 184, 198, 214, 302] [45, 82, 91, 146, 185, 199, 215, 303] [46, 83, 92, 147, 186, 200, 216, 304] [47, 84, 93, 148, 176, 201, 217, 305] [48, 85, 94, 149, 177, 202, 218, 306] [49, 86, 95, 150, 178, 203, 219, 307] [50, 87, 96, 151, 179, 204, 209, 297] [51, 77, 97, 152, 180, 205, 210, 298] [52, 78, 98, 153, 181, 206, 211, 299] [53, 79, 88, 143, 182, 207, 212, 300] [54, 80, 89, 144, 183, 208, 213, 301] [2, 55, 92, 101, 159, 195, 209, 225] [3, 56, 93, 102, 160, 196, 210, 226] [4, 57, 94, 103, 161, 197, 211, 227] [5, 58, 95, 104, 162, 187, 212, 228] [6, 59, 96, 105, 163, 188, 213, 229] [7, 60, 97, 106, 164, 189, 214, 230] [8, 61, 98, 107, 154, 190, 215, 220] [9, 62, 88, 108, 155, 191, 216, 221] [10, 63, 89, 109, 156, 192, 217, 222] [0, 64, 90, 99, 157, 193, 218, 223] [1, 65, 91, 100, 158, 194, 219, 224] [13, 66, 103, 112, 170, 206, 220, 236] [14, 67, 104, 113, 171, 207, 221, 237] [15, 68, 105, 114, 172, 208, 222, 238] [16, 69, 106, 115, 173, 198, 223, 239] [17, 70, 107, 116, 174, 199, 224, 240] [18, 71, 108, 117, 175, 200, 225, 241] [19, 72, 109, 118, 165, 201, 226, 231] [20, 73, 99, 119, 166, 202, 227, 232] [21, 74, 100, 120, 167, 203, 228, 233] [11, 75, 101, 110, 168, 204, 229, 234] [12, 76, 102, 111, 169, 205, 230, 235] [24, 77, 114, 123, 181, 217, 231, 247] [25, 78, 115, 124, 182, 218, 232, 248] [26, 79, 116, 125, 183, 219, 233, 249] [27, 80, 117, 126, 184, 209, 234, 250] [28, 81, 118, 127, 185, 210, 235, 251] [29, 82, 119, 128, 186, 211, 236, 252] [30, 83, 120, 129, 176, 212, 237, 242] [31, 84, 110, 130, 177, 213, 238, 243] [32, 85, 111, 131, 178, 214, 239, 244] [22, 86, 112, 121, 179, 215, 240, 245] [23, 87, 113, 122, 180, 216, 241, 246] [35, 88, 125, 134, 192, 228, 242, 258] [36, 89, 126, 135, 193, 229, 243, 259] [37, 90, 127, 136, 194, 230, 244, 260] [38, 91, 128, 137, 195, 220, 245, 261] [39, 92, 129, 138, 196, 221, 246, 262] [40, 93, 130, 139, 197, 222, 247, 263] [41, 94, 131, 140, 187, 223, 248, 253] [42, 95, 121, 141, 188, 224, 249, 254] [43, 96, 122, 142, 189, 225, 250, 255] [33, 97, 123, 132, 190, 226, 251, 256] [34, 98, 124, 133, 191, 227, 252, 257] [46, 99, 136, 145, 203, 239, 253, 269] [47, 100, 137, 146, 204, 240, 254, 270] [48, 101, 138, 147, 205, 241, 255, 271] [49, 102, 139, 148, 206, 231, 256, 272] [50, 103, 140, 149, 207, 232, 257, 273] [51, 104, 141, 150, 208, 233, 258, 274] [52, 105, 142, 151, 198, 234, 259, 264] [53, 106, 132, 152, 199, 235, 260, 265] [54, 107, 133, 153, 200, 236, 261, 266] [44, 108, 134, 143, 201, 237, 262, 267] [45, 109, 135, 144, 202, 238, 263, 268] [2, 57, 110, 147, 214, 250, 264, 280] [3, 58, 111, 148, 215, 251, 265, 281] [4, 59, 112, 149, 216, 252, 266, 282] [5, 60, 113, 150, 217, 242, 267, 283] [6, 61, 114, 151, 218, 243, 268, 284] [7, 62, 115, 152, 219, 244, 269, 285] [8, 63, 116, 153, 209, 245, 270, 275] [9, 64, 117, 143, 210, 246, 271, 276] [10, 65, 118, 144, 211, 247, 272, 277] [0, 55, 119, 145, 212, 248, 273, 278] [1, 56, 120, 146, 213, 249, 274, 279] [4, 13, 68, 121, 225, 261, 275, 291] [5, 14, 69, 122, 226, 262, 276, 292] [6, 15, 70, 123, 227, 263, 277, 293] [7, 16, 71, 124, 228, 253, 278, 294] [8, 17, 72, 125, 229, 254, 279, 295] [9, 18, 73, 126, 230, 255, 280, 296] [10, 19, 74, 127, 220, 256, 281, 286] [0, 20, 75, 128, 221, 257, 282, 287] [1, 21, 76, 129, 222, 258, 283, 288] [2, 11, 66, 130, 223, 259, 284, 289] [3, 12, 67, 131, 224, 260, 285, 290] [15, 24, 79, 132, 236, 272, 286, 302] [16, 25, 80, 133, 237, 273, 287, 303] [17, 26, 81, 134, 238, 274, 288, 304] [18, 27, 82, 135, 239, 264, 289, 305] [19, 28, 83, 136, 240, 265, 290, 306] [20, 29, 84, 137, 241, 266, 291, 307] [21, 30, 85, 138, 231, 267, 292, 297] [11, 31, 86, 139, 232, 268, 293, 298] [12, 32, 87, 140, 233, 269, 294, 299] [13, 22, 77, 141, 234, 270, 295, 300] [14, 23, 78, 142, 235, 271, 296, 301] [26, 35, 90, 143, 159, 247, 283, 297] [27, 36, 91, 144, 160, 248, 284, 298] [28, 37, 92, 145, 161, 249, 285, 299] [29, 38, 93, 146, 162, 250, 275, 300] [30, 39, 94, 147, 163, 251, 276, 301] [31, 40, 95, 148, 164, 252, 277, 302] [32, 41, 96, 149, 154, 242, 278, 303] [22, 42, 97, 150, 155, 243, 279, 304] [23, 43, 98, 151, 156, 244, 280, 305] [24, 33, 88, 152, 157, 245, 281, 306] [25, 34, 89, 153, 158, 246, 282, 307]
Code ID 308-10-26 · download JSON · raw on GitHub