[[184,50,10]] · QEC Challenge

[[184,50,10]] d =

184

kd²/n

27.174

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[11, 14, 22, 24, 29, 33, 79, 101, 163, 167]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[25, 29, 106, 108, 117, 118, 121, 124, 149, 151]

certificate exact, d = 10 · CryptoMiniSat 5.14.7 SAT

X: no logical < 10 exists; Z: no logical < 10 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 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 3 · H_Z 3

trapping sets H_X (1,3)×184 (2,4)×1932 (3,3)×161 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 184 (2,4): 1932 (3,3): 161 (3,5): 26565 (3,7): 3864

trapping sets H_Z (1,3)×184 (2,4)×1932 (3,3)×161 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 184 (2,4): 1932 (3,3): 161 (3,5): 26565 (3,7): 3864

Construction & provenance

authors Lee, Jong Yeon

provenance literature baseline

construction Cyclic all-one pair-partition CPM CSS construction (arXiv:2607.14091), contributed instance PPS184 (internal id j3l8_p23_00156) from the Okada-Kasai catalogue's contributed-codes section: lift size p=23, (J,L)=(3,8), n=8p=184, 3p=69 checks per side, column weight 3, row weight 8, Tanner girth 6. CPM exponent arrays (rows = the three CPM block-rows per side, columns = the 8 block-columns): E_X = [[0,0,0,0,0,0,0,0],[0,12,8,21,6,1,19,15],[0,9,18,11,7,17,10,4]], E_Z = [[0,15,7,22,7,0,22,15],[0,1,2,4,0,1,2,4],[0,5,17,6,6,17,5,0]]. Lift convention per the authors' expand_and_verify_cpm_matrices.py (github.com/kasaikenta/pair-partition-cpm-css-codes @ 9c2a6f2): check a of block-row b on side S has support {8*((a - E_S[b][t]) mod 23) + t : t=0..7}. Source file https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/contributed_codes/jong_yeon_lee/codes/PPS184_n184_k50_d10.cpm, expanded with the authors' own script.

model classical construction (no AI model)

reference arXiv:2607.14091

date 2026

notes Literature baseline, seeded 2026-08-14. Listed as PPS184 [[184,50,10]] in the contributed-codes section (Jong Yeon Lee, 19 CPM-PP codes) of the Okada-Kasai pair-partition CPM catalogue (arXiv:2607.14091; catalogue page updated 2026-08-13), which reports the distance as exact. Unlike the catalogue's primary instances, no public machine-verified distance certificate bundle covers this code (checked master_manifest.json of certificate_bundles_20260805), so it is recorded here strictly as a witness-backed upper bound: d<=10 on both sides from this repo's own 150k-trial RIS search (seed 7). Reproduction was validated end to end: the published .cpm file expanded with the authors' own script gives a CSS-orthogonal (3,8)-regular pair with GF(2) ranks 67/67 (k=50). Best weight-8 kd^2/n (27.17) below n=472 on the board, and the highest k of any weight-8 entry below n=472.

family pair-partition CPM (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 69 (max weight 8) · Z-checks 69 (max weight 8)

H_X (69 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] [0, 19, 38, 71, 89, 122, 140, 181] [5, 8, 27, 46, 79, 97, 130, 148] [13, 16, 35, 54, 87, 105, 138, 156] [21, 24, 43, 62, 95, 113, 146, 164] [29, 32, 51, 70, 103, 121, 154, 172] [37, 40, 59, 78, 111, 129, 162, 180] [4, 45, 48, 67, 86, 119, 137, 170] [12, 53, 56, 75, 94, 127, 145, 178] [2, 20, 61, 64, 83, 102, 135, 153] [10, 28, 69, 72, 91, 110, 143, 161] [18, 36, 77, 80, 99, 118, 151, 169] [26, 44, 85, 88, 107, 126, 159, 177] [1, 34, 52, 93, 96, 115, 134, 167] [9, 42, 60, 101, 104, 123, 142, 175] [17, 50, 68, 109, 112, 131, 150, 183] [7, 25, 58, 76, 117, 120, 139, 158] [15, 33, 66, 84, 125, 128, 147, 166] [23, 41, 74, 92, 133, 136, 155, 174] [31, 49, 82, 100, 141, 144, 163, 182] [6, 39, 57, 90, 108, 149, 152, 171] [14, 47, 65, 98, 116, 157, 160, 179] [3, 22, 55, 73, 106, 124, 165, 168] [11, 30, 63, 81, 114, 132, 173, 176] [0, 42, 53, 99, 110, 113, 132, 159] [8, 50, 61, 107, 118, 121, 140, 167] [16, 58, 69, 115, 126, 129, 148, 175] [24, 66, 77, 123, 134, 137, 156, 183] [7, 32, 74, 85, 131, 142, 145, 164] [15, 40, 82, 93, 139, 150, 153, 172] [23, 48, 90, 101, 147, 158, 161, 180] [4, 31, 56, 98, 109, 155, 166, 169] [12, 39, 64, 106, 117, 163, 174, 177] [1, 20, 47, 72, 114, 125, 171, 182] [6, 9, 28, 55, 80, 122, 133, 179] [3, 14, 17, 36, 63, 88, 130, 141] [11, 22, 25, 44, 71, 96, 138, 149] [19, 30, 33, 52, 79, 104, 146, 157] [27, 38, 41, 60, 87, 112, 154, 165] [35, 46, 49, 68, 95, 120, 162, 173] [43, 54, 57, 76, 103, 128, 170, 181] [5, 51, 62, 65, 84, 111, 136, 178] [2, 13, 59, 70, 73, 92, 119, 144] [10, 21, 67, 78, 81, 100, 127, 152] [18, 29, 75, 86, 89, 108, 135, 160] [26, 37, 83, 94, 97, 116, 143, 168] [34, 45, 91, 102, 105, 124, 151, 176]

H_Z (69 checks, sparse supports)

[0, 5, 11, 14, 65, 71, 130, 132] [8, 13, 19, 22, 73, 79, 138, 140] [16, 21, 27, 30, 81, 87, 146, 148] [24, 29, 35, 38, 89, 95, 154, 156] [32, 37, 43, 46, 97, 103, 162, 164] [40, 45, 51, 54, 105, 111, 170, 172] [48, 53, 59, 62, 113, 119, 178, 180] [2, 4, 56, 61, 67, 70, 121, 127] [10, 12, 64, 69, 75, 78, 129, 135] [18, 20, 72, 77, 83, 86, 137, 143] [26, 28, 80, 85, 91, 94, 145, 151] [34, 36, 88, 93, 99, 102, 153, 159] [42, 44, 96, 101, 107, 110, 161, 167] [50, 52, 104, 109, 115, 118, 169, 175] [58, 60, 112, 117, 123, 126, 177, 183] [1, 7, 66, 68, 120, 125, 131, 134] [9, 15, 74, 76, 128, 133, 139, 142] [17, 23, 82, 84, 136, 141, 147, 150] [25, 31, 90, 92, 144, 149, 155, 158] [33, 39, 98, 100, 152, 157, 163, 166] [41, 47, 106, 108, 160, 165, 171, 174] [49, 55, 114, 116, 168, 173, 179, 182] [3, 6, 57, 63, 122, 124, 176, 181] [0, 4, 155, 159, 170, 174, 177, 181] [1, 5, 8, 12, 163, 167, 178, 182] [2, 6, 9, 13, 16, 20, 171, 175] [10, 14, 17, 21, 24, 28, 179, 183] [3, 7, 18, 22, 25, 29, 32, 36] [11, 15, 26, 30, 33, 37, 40, 44] [19, 23, 34, 38, 41, 45, 48, 52] [27, 31, 42, 46, 49, 53, 56, 60] [35, 39, 50, 54, 57, 61, 64, 68] [43, 47, 58, 62, 65, 69, 72, 76] [51, 55, 66, 70, 73, 77, 80, 84] [59, 63, 74, 78, 81, 85, 88, 92] [67, 71, 82, 86, 89, 93, 96, 100] [75, 79, 90, 94, 97, 101, 104, 108] [83, 87, 98, 102, 105, 109, 112, 116] [91, 95, 106, 110, 113, 117, 120, 124] [99, 103, 114, 118, 121, 125, 128, 132] [107, 111, 122, 126, 129, 133, 136, 140] [115, 119, 130, 134, 137, 141, 144, 148] [123, 127, 138, 142, 145, 149, 152, 156] [131, 135, 146, 150, 153, 157, 160, 164] [139, 143, 154, 158, 161, 165, 168, 172] [147, 151, 162, 166, 169, 173, 176, 180] [0, 7, 50, 53, 139, 140, 145, 150] [8, 15, 58, 61, 147, 148, 153, 158] [16, 23, 66, 69, 155, 156, 161, 166] [24, 31, 74, 77, 163, 164, 169, 174] [32, 39, 82, 85, 171, 172, 177, 182] [1, 6, 40, 47, 90, 93, 179, 180] [3, 4, 9, 14, 48, 55, 98, 101] [11, 12, 17, 22, 56, 63, 106, 109] [19, 20, 25, 30, 64, 71, 114, 117] [27, 28, 33, 38, 72, 79, 122, 125] [35, 36, 41, 46, 80, 87, 130, 133] [43, 44, 49, 54, 88, 95, 138, 141] [51, 52, 57, 62, 96, 103, 146, 149] [59, 60, 65, 70, 104, 111, 154, 157] [67, 68, 73, 78, 112, 119, 162, 165] [75, 76, 81, 86, 120, 127, 170, 173] [83, 84, 89, 94, 128, 135, 178, 181] [2, 5, 91, 92, 97, 102, 136, 143] [10, 13, 99, 100, 105, 110, 144, 151] [18, 21, 107, 108, 113, 118, 152, 159] [26, 29, 115, 116, 121, 126, 160, 167] [34, 37, 123, 124, 129, 134, 168, 175] [42, 45, 131, 132, 137, 142, 176, 183]

Code ID 184-50-10 · download JSON · raw on GitHub