[[148,44,4]] · QEC Challenge

[[148,44,4]] d =

148

kd²/n

4.757

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 20, w_Z = 13 (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 operator (support, 4 qubits)

[73, 76, 78, 79]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[33, 73, 113, 121]

certificate exact, d = 4 · scipy/HiGHS MILP

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 9–20 (mean 12.1) · H_Z 5–13 (mean 8.0)

qubit degrees H_X 1–8 (mean 3.27) · H_Z 1–16 (mean 3.459)

trapping sets H_X (1,1)×8 (2,1)×256 (3,1)×280 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 32 (1,3): 48 (1,4): 47 (1,5): 8 (1,8): 5 (2,1): 256 (2,2): 384 (2,3): 268 (2,4): 190 (2,5): 216 (2,6): 192 (2,7): 68 (2,8): 64 (2,9): 144 (2,10): 128 (2,11): 40 (3,1): 280 (3,2): 1138 (3,3): 2232 (3,4): 3060 (3,5): 4282 (3,6): 4528 (3,7): 4380 (3,8): 4208 (3,9): 4554 (3,10): 4850 (3,11): 4624 (3,12): 3104 (3,13): 1136 (3,14): 172 (3,15): 144 (3,16): 192 (3,17): 80

trapping sets H_Z (1,1)×16 (2,1)×192 (3,1)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 16 (1,2): 48 (1,3): 48 (1,4): 16 (1,8): 16 (1,16): 4 (2,1): 192 (2,2): 192 (2,3): 64 (2,7): 32 (2,8): 232 (2,9): 288 (2,10): 128 (2,15): 16 (2,16): 96 (2,17): 144 (2,18): 64 (3,1): 112 (3,2): 336 (3,3): 336 (3,4): 112 (3,6): 112 (3,7): 1136 (3,8): 3304 (3,9): 5512 (3,10): 4928 (3,11): 2648 (3,12): 672 (3,14): 360 (3,15): 1632 (3,16): 3024 (3,17): 4416 (3,18): 4248 (3,19): 2224 (3,20): 480 (3,22): 192 (3,23): 576 (3,24): 384 (3,30): 48 (3,31): 144 (3,32): 96

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Hypergraph product of classical codes eham16 and eham8

model Tencent Hy3 (claimed, not verified)

builds on arXiv:0903.0566

date 2026-07-17

family hypergraph product (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight > 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 40 (max weight 20) · Z-checks 64 (max weight 13)

H_X (40 checks, sparse supports)

[0, 16, 32, 48, 64, 80, 96, 112, 128, 131] [1, 17, 33, 49, 65, 81, 97, 113, 129, 131] [2, 18, 34, 50, 66, 82, 98, 114, 128, 129, 131] [3, 19, 35, 51, 67, 83, 99, 115, 130, 131] [4, 20, 36, 52, 68, 84, 100, 116, 128, 130, 131] [5, 21, 37, 53, 69, 85, 101, 117, 129, 130, 131] [6, 22, 38, 54, 70, 86, 102, 118, 128, 129, 130, 131] [7, 23, 39, 55, 71, 87, 103, 119, 131] [8, 16, 40, 48, 72, 80, 104, 112, 132, 135] [9, 17, 41, 49, 73, 81, 105, 113, 133, 135] [10, 18, 42, 50, 74, 82, 106, 114, 132, 133, 135] [11, 19, 43, 51, 75, 83, 107, 115, 134, 135] [12, 20, 44, 52, 76, 84, 108, 116, 132, 134, 135] [13, 21, 45, 53, 77, 85, 109, 117, 133, 134, 135] [14, 22, 46, 54, 78, 86, 110, 118, 132, 133, 134, 135] [15, 23, 47, 55, 79, 87, 111, 119, 135] [24, 32, 40, 48, 88, 96, 104, 112, 136, 139] [25, 33, 41, 49, 89, 97, 105, 113, 137, 139] [26, 34, 42, 50, 90, 98, 106, 114, 136, 137, 139] [27, 35, 43, 51, 91, 99, 107, 115, 138, 139] [28, 36, 44, 52, 92, 100, 108, 116, 136, 138, 139] [29, 37, 45, 53, 93, 101, 109, 117, 137, 138, 139] [30, 38, 46, 54, 94, 102, 110, 118, 136, 137, 138, 139] [31, 39, 47, 55, 95, 103, 111, 119, 139] [56, 64, 72, 80, 88, 96, 104, 112, 140, 143] [57, 65, 73, 81, 89, 97, 105, 113, 141, 143] [58, 66, 74, 82, 90, 98, 106, 114, 140, 141, 143] [59, 67, 75, 83, 91, 99, 107, 115, 142, 143] [60, 68, 76, 84, 92, 100, 108, 116, 140, 142, 143] [61, 69, 77, 85, 93, 101, 109, 117, 141, 142, 143] [62, 70, 78, 86, 94, 102, 110, 118, 140, 141, 142, 143] [63, 71, 79, 87, 95, 103, 111, 119, 143] [0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 144, 147] [1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121, 145, 147] [2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122, 144, 145, 147] [3, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 99, 107, 115, 123, 146, 147] [4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 144, 146, 147] [5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109, 117, 125, 145, 146, 147] [6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 144, 145, 146, 147] [7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 147]

H_Z (64 checks, sparse supports)

[0, 2, 4, 6, 128, 144] [1, 2, 5, 6, 129, 145] [3, 4, 5, 6, 130, 146] [0, 1, 2, 3, 4, 5, 6, 7, 131, 147] [8, 10, 12, 14, 132, 144] [9, 10, 13, 14, 133, 145] [11, 12, 13, 14, 134, 146] [8, 9, 10, 11, 12, 13, 14, 15, 135, 147] [16, 18, 20, 22, 128, 132, 144] [17, 18, 21, 22, 129, 133, 145] [19, 20, 21, 22, 130, 134, 146] [16, 17, 18, 19, 20, 21, 22, 23, 131, 135, 147] [24, 26, 28, 30, 136, 144] [25, 26, 29, 30, 137, 145] [27, 28, 29, 30, 138, 146] [24, 25, 26, 27, 28, 29, 30, 31, 139, 147] [32, 34, 36, 38, 128, 136, 144] [33, 34, 37, 38, 129, 137, 145] [35, 36, 37, 38, 130, 138, 146] [32, 33, 34, 35, 36, 37, 38, 39, 131, 139, 147] [40, 42, 44, 46, 132, 136, 144] [41, 42, 45, 46, 133, 137, 145] [43, 44, 45, 46, 134, 138, 146] [40, 41, 42, 43, 44, 45, 46, 47, 135, 139, 147] [48, 50, 52, 54, 128, 132, 136, 144] [49, 50, 53, 54, 129, 133, 137, 145] [51, 52, 53, 54, 130, 134, 138, 146] [48, 49, 50, 51, 52, 53, 54, 55, 131, 135, 139, 147] [56, 58, 60, 62, 140, 144] [57, 58, 61, 62, 141, 145] [59, 60, 61, 62, 142, 146] [56, 57, 58, 59, 60, 61, 62, 63, 143, 147] [64, 66, 68, 70, 128, 140, 144] [65, 66, 69, 70, 129, 141, 145] [67, 68, 69, 70, 130, 142, 146] [64, 65, 66, 67, 68, 69, 70, 71, 131, 143, 147] [72, 74, 76, 78, 132, 140, 144] [73, 74, 77, 78, 133, 141, 145] [75, 76, 77, 78, 134, 142, 146] [72, 73, 74, 75, 76, 77, 78, 79, 135, 143, 147] [80, 82, 84, 86, 128, 132, 140, 144] [81, 82, 85, 86, 129, 133, 141, 145] [83, 84, 85, 86, 130, 134, 142, 146] [80, 81, 82, 83, 84, 85, 86, 87, 131, 135, 143, 147] [88, 90, 92, 94, 136, 140, 144] [89, 90, 93, 94, 137, 141, 145] [91, 92, 93, 94, 138, 142, 146] [88, 89, 90, 91, 92, 93, 94, 95, 139, 143, 147] [96, 98, 100, 102, 128, 136, 140, 144] [97, 98, 101, 102, 129, 137, 141, 145] [99, 100, 101, 102, 130, 138, 142, 146] [96, 97, 98, 99, 100, 101, 102, 103, 131, 139, 143, 147] [104, 106, 108, 110, 132, 136, 140, 144] [105, 106, 109, 110, 133, 137, 141, 145] [107, 108, 109, 110, 134, 138, 142, 146] [104, 105, 106, 107, 108, 109, 110, 111, 135, 139, 143, 147] [112, 114, 116, 118, 128, 132, 136, 140, 144] [113, 114, 117, 118, 129, 133, 137, 141, 145] [115, 116, 117, 118, 130, 134, 138, 142, 146] [112, 113, 114, 115, 116, 117, 118, 119, 131, 135, 139, 143, 147] [120, 122, 124, 126, 144] [121, 122, 125, 126, 145] [123, 124, 125, 126, 146] [120, 121, 122, 123, 124, 125, 126, 127, 147]

Code ID 148-44-4 · download JSON · raw on GitHub