[[136,38,8]] · QEC Challenge

[[136,38,8]] d =

136

kd²/n

17.882

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[34, 41, 48, 64, 83, 99, 106, 113]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[65, 66, 75, 77, 90, 94, 113, 119]

certificate exact, d = 8 · CryptoMiniSat 5.14.7 SAT

X: no logical < 8 exists; Z: no logical < 8 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)×136 (2,4)×1428 (3,3)×340 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 136 (2,4): 1428 (3,3): 340 (3,5): 18972 (3,7): 2856

trapping sets H_Z (1,3)×136 (2,4)×1428 (3,3)×340 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 136 (2,4): 1428 (3,3): 340 (3,5): 18972 (3,7): 2856

Construction & provenance

authors Lee, Jong Yeon

provenance literature baseline

construction Cyclic all-one pair-partition CPM CSS construction (arXiv:2607.14091), instance PPS136_n136_k38_d8 (internal id j3l8_p17_00010): lift size p=17, (J,L)=(3,8), n=8*p=136, 3*p=51 checks per side, column weight 3, row weight 8. CPM exponent arrays (rows = CPM block-rows, columns = the 8 block-columns): E_X = [[0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 8, 10, 4, 1, 9, 6], [0, 16, 13, 12, 7, 10, 2, 5]], E_Z = [[0, 6, 7, 13, 7, 0, 13, 6], [0, 1, 4, 5, 0, 1, 4, 5], [0, 11, 10, 4, 4, 10, 11, 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 17) + 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/PPS136_n136_k38_d8.cpm (sha256 2e91f898d06edf32773bc3a37129f17b2309ced5e24d21b384043d5d2740b2e3), 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 PPS136 [[136,38,8]] 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. No public machine-verified distance certificate bundle covers this instance (checked master_manifest.json of certificate_bundles_20260805), so it is recorded here strictly as a witness-backed upper bound from this repo's own 200000-trial RIS search per side (seed 11). Reproduction was validated end to end: the published .cpm file expanded with the authors' own script gives a CSS-orthogonal regular pair with the claimed k.

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 51 (max weight 8) · Z-checks 51 (max weight 8)

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

H_Z (51 checks, sparse supports)

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

Code ID 136-38-8 · download JSON · raw on GitHub