[[224,12,21]] · QEC Challenge

[[224,12,21]] d ≤

224

kd²/n

23.625

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[32, 54, 74, 77, 87, 88, 90, 98, 99, 114, 119, 133, 134, 139, 147, 149, 152, 154, 168, 187, 209]

d_Z 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[10, 29, 32, 62, 63, 76, 80, 93, 121, 122, 137, 142, 150, 164, 167, 170, 184, 187, 191, 200, 220]

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

check weights H_X 9 · H_Z 9

qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)

trapping sets H_X (1,4)×112 (2,5)×140 (3,4)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 112 (1,5): 112 (2,5): 140 (2,6): 728 (2,7): 1960 (2,8): 1008 (3,4): 28 (3,5): 28 (3,6): 644 (3,7): 3164 (3,8): 9884 (3,9): 32984 (3,10): 38388 (3,11): 12544 (3,12): 2996 (3,13): 784

trapping sets H_Z (1,4)×112 (2,5)×140 (3,4)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 112 (1,5): 112 (2,5): 140 (2,6): 728 (2,7): 1960 (2,8): 1008 (3,4): 28 (3,5): 28 (3,6): 644 (3,7): 3164 (3,8): 9884 (3,9): 32984 (3,10): 38388 (3,11): 12544 (3,12): 2996 (3,13): 784

Construction & provenance

authors @rajendrarajpurohit

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction Coset lifted product (CLP) code, Definition III.1 of arXiv:2610.12782, 1x1 over 2x2 layout, check weight 9. G = SmallGroup(224,169) (GAP SmallGroups id), H = <f2, f6> = C4 by generators in the polycyclic generating sequence f1,f2,... of G, m = [G:H] = 56, n = 4m = 224. a over F_2[G] by the left coset action on G/H, B over F_2[N_G(H)] by the right coset action. a = 1 + f1*f4 + f1*f3*f4² + f2*f4²*f5*f6 + f1*f3*f4⁵*f5*f6; B = [[1, 1 + f3*f4³*f5 + f3*f4⁵], [1 + f4⁴*f5 + f3*f4⁵, f5]]. Rebuild with gap/rebuild.g in https://github.com/rajendrarajpurohit/CLP_QLDPC.

model human (claimed, not verified)

builds on arXiv:2610.12782

date 2026-11-03

notes Table II of arXiv:2610.12782 lists this code with the two sided QDistRnd bound d <= 21 at 10^6 rounds; the witness here is from this tool's own search and the entry carries that value. Authors Rajendra Rajpurohit, V. Lalitha, Shayan Srinivasa Garani.

family 2BGA coset (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

[[224,12,21]] CLP code, check weight 9

Primary reference: arXiv:2610.12782, *Coset Lifted Product Codes*, Rajpurohit, Lalitha, Garani, Table II. Data and scripts: https://github.com/rajendrarajpurohit/CLP_QLDPC.

Direction and hypothesis

Part of the paper's search over 14,843 pairs (G,H), 30 <= m <= 100, at check weights 7 to 10, for codes with kd^2/n above the published bar at each weight. This code is one of the paper's Table II candidates, whose distances are upper bounds only; the paper's integer program certification does not reach d > 20.

This code

G = SmallGroup(224,169), H = <f2, f6> = C4, m = 56, n = 4m, check weight 9. Protographs a and B in the entry's construction field.

Evidence trail

| stage | rounds / trials | result | |---|---|---| | search log | 2 x 10^5 QDistRnd per side | d <= 21 | | rescreen | 10^6 QDistRnd per side, both sides | d <= 21 | | this tool | 20,000 RIS trials per side | X <= 21, Z <= 21 |

Not certified exact. The witnesses in the entry are the ones this tool found, checked by the verifier.

Dead ends

In the paper's screen, six of seven candidates with bounds at or above 25 lost weight between 2 x 10^5 and 10^6 rounds; this one held. Non square layouts and 3x3 shapes with single element entries were closed as negative results in the paper.

Tools

GAP 4.15.1 with SmallGroups and QDistRnd, the construction and driver scripts of the repository above, no LLM in the search loop. Witness and verification: this repository's qldpc submit.

Reproduction

gap/rebuild.g in the repository above, with G, H, a, B as given, rebuilds HX and HZ.

Parity checks

X-checks 112 (max weight 9) · Z-checks 112 (max weight 9)

H_X (112 checks, sparse supports)

[0, 6, 17, 18, 53, 112, 168, 199, 206] [1, 3, 12, 14, 47, 113, 169, 210, 211] [2, 11, 12, 23, 51, 114, 170, 193, 200] [3, 12, 25, 26, 55, 115, 171, 207, 214] [4, 9, 21, 35, 42, 116, 172, 190, 215] [5, 7, 9, 18, 41, 117, 173, 204, 205] [6, 9, 20, 22, 52, 118, 174, 217, 218] [7, 18, 20, 31, 54, 119, 175, 201, 208] [8, 14, 27, 29, 36, 120, 176, 184, 209] [9, 19, 20, 33, 34, 121, 170, 177, 215] [10, 16, 29, 43, 49, 122, 178, 198, 220] [11, 14, 16, 26, 48, 123, 179, 212, 213] [8, 12, 16, 28, 30, 124, 173, 180, 221] [10, 13, 23, 29, 38, 125, 181, 194, 221] [13, 14, 26, 28, 39, 126, 168, 182, 209] [15, 22, 35, 37, 44, 127, 183, 192, 216] [16, 27, 28, 41, 42, 128, 175, 184, 220] [5, 17, 24, 37, 50, 129, 176, 185, 206] [4, 18, 22, 24, 34, 130, 169, 186, 219] [15, 17, 19, 32, 35, 131, 187, 188, 219] [15, 20, 24, 36, 38, 132, 179, 188, 223] [17, 21, 31, 37, 46, 133, 189, 202, 223] [21, 22, 34, 36, 47, 134, 171, 190, 216] [1, 23, 30, 43, 45, 135, 172, 191, 200] [24, 35, 36, 48, 49, 136, 176, 182, 192] [11, 25, 32, 45, 53, 137, 183, 193, 214] [10, 26, 30, 32, 42, 138, 174, 194, 222] [23, 25, 27, 40, 43, 139, 195, 196, 222] [23, 28, 32, 44, 46, 140, 186, 187, 196] [2, 25, 29, 39, 45, 141, 187, 197, 210] [29, 30, 42, 44, 52, 142, 172, 177, 198] [6, 31, 38, 50, 51, 143, 178, 199, 208] [4, 5, 32, 43, 44, 144, 183, 190, 200] [18, 33, 40, 51, 55, 145, 170, 191, 201] [17, 34, 38, 40, 49, 146, 180, 181, 202] [0, 31, 33, 35, 50, 147, 181, 203, 204] [1, 2, 31, 36, 40, 148, 194, 195, 204] [7, 33, 37, 47, 51, 149, 195, 205, 217] [1, 8, 37, 38, 49, 150, 178, 184, 206] [12, 39, 46, 53, 54, 151, 168, 185, 207] [1, 10, 11, 40, 50, 152, 191, 198, 208] [0, 19, 26, 41, 54, 153, 175, 199, 209] [0, 5, 25, 42, 46, 154, 188, 189, 210] [3, 39, 41, 43, 53, 155, 189, 211, 212] [0, 6, 7, 39, 44, 156, 202, 203, 212] [14, 41, 45, 52, 54, 157, 173, 203, 213] [5, 6, 15, 45, 46, 158, 185, 192, 214] [2, 13, 20, 47, 55, 159, 171, 193, 215] [3, 13, 27, 34, 48, 160, 182, 207, 216] [2, 3, 11, 33, 49, 161, 196, 197, 217] [9, 47, 48, 50, 55, 162, 169, 197, 218] [8, 13, 22, 48, 51, 163, 179, 211, 219] [7, 19, 21, 28, 52, 164, 177, 201, 220] [4, 16, 19, 52, 53, 165, 174, 205, 221] [4, 15, 21, 30, 54, 166, 186, 218, 222] [8, 10, 24, 27, 55, 167, 180, 213, 223] [56, 62, 73, 74, 109, 112, 145, 150, 172] [57, 59, 68, 70, 103, 113, 154, 157, 181] [58, 67, 68, 79, 107, 114, 144, 151, 176] [59, 68, 81, 82, 111, 115, 153, 158, 178] [60, 65, 77, 91, 98, 116, 136, 159, 168] [61, 63, 65, 74, 97, 117, 148, 162, 187] [62, 65, 76, 78, 108, 118, 161, 163, 189] [63, 74, 76, 87, 110, 119, 152, 159, 183] [64, 70, 83, 85, 92, 120, 142, 153, 170] [65, 75, 76, 89, 90, 114, 121, 160, 185] [66, 72, 85, 99, 105, 122, 144, 164, 171] [67, 70, 72, 82, 104, 123, 156, 165, 195] [64, 68, 72, 84, 86, 117, 124, 166, 197] [66, 69, 79, 85, 94, 125, 140, 165, 169] [69, 70, 82, 84, 95, 112, 126, 164, 191] [71, 78, 91, 93, 100, 127, 150, 160, 175] [72, 83, 84, 97, 98, 116, 119, 128, 193] [61, 73, 80, 93, 106, 120, 129, 152, 177] [60, 74, 78, 80, 90, 113, 130, 167, 203] [71, 73, 75, 88, 91, 131, 146, 163, 173] [71, 76, 80, 92, 94, 123, 125, 132, 205] [73, 77, 87, 93, 102, 133, 148, 167, 174] [77, 78, 90, 92, 103, 115, 120, 134, 199] [57, 79, 86, 99, 101, 116, 135, 158, 182] [80, 91, 92, 104, 105, 122, 126, 136, 201] [67, 81, 88, 101, 109, 112, 127, 137, 184] [66, 82, 86, 88, 98, 118, 131, 138, 211] [79, 81, 83, 96, 99, 139, 154, 166, 179] [79, 84, 88, 100, 102, 130, 133, 140, 213] [58, 81, 85, 95, 101, 131, 141, 156, 180] [85, 86, 98, 100, 108, 121, 127, 142, 207] [62, 87, 94, 106, 107, 114, 122, 143, 190] [60, 61, 88, 99, 100, 129, 134, 144, 209] [74, 89, 96, 107, 111, 115, 135, 145, 192] [73, 90, 94, 96, 105, 124, 139, 146, 218] [56, 87, 89, 91, 106, 125, 147, 161, 186] [57, 58, 87, 92, 96, 138, 141, 148, 219] [63, 89, 93, 103, 107, 113, 139, 149, 188] [57, 64, 93, 94, 105, 128, 135, 150, 215] [68, 95, 102, 109, 110, 119, 129, 151, 198] [57, 66, 67, 96, 106, 137, 142, 152, 216] [56, 75, 82, 97, 110, 121, 143, 153, 200] [56, 61, 81, 98, 102, 132, 147, 154, 221] [59, 95, 97, 99, 109, 117, 133, 155, 194] [56, 62, 63, 95, 100, 146, 149, 156, 222] [70, 97, 101, 108, 110, 118, 147, 157, 196] [61, 62, 71, 101, 102, 136, 143, 158, 220] [58, 69, 76, 103, 111, 126, 137, 159, 206] [59, 69, 83, 90, 104, 128, 151, 160, 208] [58, 59, 67, 89, 105, 140, 155, 161, 223] [65, 103, 104, 106, 111, 123, 141, 162, 202] [64, 69, 78, 104, 107, 124, 155, 163, 204] [63, 75, 77, 84, 108, 134, 145, 164, 214] [60, 72, 75, 108, 109, 130, 149, 165, 210] [60, 71, 77, 86, 110, 132, 162, 166, 212] [64, 66, 80, 83, 111, 138, 157, 167, 217]

H_Z (112 checks, sparse supports)

[0, 56, 70, 81, 112, 147, 153, 154, 156] [1, 57, 74, 93, 113, 135, 148, 150, 152] [2, 58, 65, 87, 114, 141, 148, 159, 161] [3, 59, 78, 89, 113, 115, 155, 160, 161] [4, 60, 72, 79, 116, 130, 144, 165, 166] [5, 61, 68, 99, 117, 129, 144, 154, 158] [6, 62, 82, 101, 112, 118, 143, 156, 158] [7, 63, 72, 95, 117, 119, 149, 156, 164] [8, 64, 73, 78, 120, 124, 150, 163, 167] [9, 65, 86, 97, 116, 117, 118, 121, 162] [10, 66, 80, 87, 122, 125, 138, 152, 167] [11, 67, 76, 106, 114, 123, 137, 152, 161] [12, 68, 90, 107, 113, 114, 115, 124, 151] [13, 69, 76, 91, 125, 126, 159, 160, 163] [14, 70, 80, 103, 113, 120, 123, 126, 157] [15, 71, 81, 86, 127, 131, 132, 158, 166] [16, 72, 94, 104, 122, 123, 124, 128, 165] [17, 73, 88, 95, 112, 129, 131, 133, 146] [18, 74, 84, 109, 112, 117, 119, 130, 145] [19, 75, 82, 85, 121, 131, 153, 164, 165] [20, 76, 98, 110, 118, 119, 121, 132, 159] [21, 77, 84, 99, 116, 133, 134, 164, 166] [22, 78, 88, 108, 118, 127, 130, 134, 163] [23, 79, 89, 94, 114, 125, 135, 139, 140] [24, 60, 80, 102, 129, 130, 132, 136, 167] [25, 81, 96, 103, 115, 137, 139, 141, 154] [26, 82, 92, 111, 115, 123, 126, 138, 153] [27, 83, 90, 93, 120, 128, 139, 160, 167] [28, 69, 84, 105, 124, 126, 128, 140, 164] [29, 85, 92, 106, 120, 122, 125, 141, 142] [30, 64, 86, 96, 124, 135, 138, 142, 166] [31, 87, 97, 102, 119, 133, 143, 147, 148] [32, 58, 66, 88, 131, 137, 138, 140, 144] [33, 56, 89, 108, 121, 145, 147, 149, 161] [34, 75, 90, 100, 121, 130, 134, 146, 160] [35, 91, 98, 101, 116, 127, 131, 136, 147] [36, 61, 77, 92, 120, 132, 134, 136, 148] [37, 93, 100, 109, 127, 129, 133, 149, 150] [38, 56, 71, 94, 125, 132, 143, 146, 150] [39, 58, 95, 104, 126, 141, 151, 155, 156] [40, 63, 73, 96, 139, 145, 146, 148, 152] [41, 59, 64, 97, 117, 128, 153, 155, 157] [42, 57, 83, 98, 116, 128, 138, 142, 154] [43, 99, 105, 107, 122, 135, 139, 144, 155] [44, 67, 85, 100, 127, 140, 142, 144, 156] [45, 57, 101, 111, 135, 137, 141, 157, 158] [46, 59, 79, 102, 133, 140, 151, 154, 158] [47, 60, 63, 103, 113, 134, 149, 159, 162] [48, 65, 71, 104, 123, 136, 160, 162, 163] [49, 62, 91, 105, 122, 136, 146, 150, 161] [50, 61, 106, 110, 129, 143, 147, 152, 162] [51, 62, 75, 107, 114, 143, 145, 149, 163] [52, 66, 70, 108, 118, 142, 157, 164, 165] [53, 67, 69, 109, 112, 137, 151, 155, 165] [54, 68, 83, 110, 119, 151, 153, 157, 166] [55, 74, 77, 111, 115, 145, 159, 162, 167] [0, 14, 39, 60, 168, 203, 209, 210, 212] [1, 18, 50, 69, 169, 191, 204, 206, 208] [2, 9, 33, 64, 170, 197, 204, 215, 217] [3, 22, 47, 66, 169, 171, 211, 216, 217] [4, 23, 30, 56, 172, 186, 200, 221, 222] [5, 12, 45, 75, 173, 185, 200, 210, 214] [6, 26, 53, 77, 168, 174, 199, 212, 214] [7, 16, 41, 71, 173, 175, 205, 212, 220] [8, 17, 24, 58, 176, 180, 206, 219, 223] [9, 30, 52, 73, 172, 173, 174, 177, 218] [10, 31, 38, 59, 178, 181, 194, 208, 223] [11, 20, 51, 83, 170, 179, 193, 208, 217] [12, 34, 55, 85, 169, 170, 171, 180, 207] [13, 34, 35, 57, 181, 182, 215, 216, 219] [14, 24, 48, 79, 169, 176, 179, 182, 213] [15, 25, 32, 63, 183, 187, 188, 214, 222] [8, 16, 38, 81, 178, 179, 180, 184, 221] [17, 39, 46, 65, 168, 185, 187, 189, 202] [18, 28, 54, 91, 168, 173, 175, 186, 201] [19, 28, 29, 61, 177, 187, 209, 220, 221] [19, 20, 42, 93, 174, 175, 177, 188, 215] [21, 42, 43, 62, 172, 189, 190, 220, 222] [4, 22, 32, 87, 174, 183, 186, 190, 219] [23, 33, 40, 70, 170, 181, 191, 195, 196] [15, 24, 46, 89, 185, 186, 188, 192, 223] [2, 25, 47, 72, 171, 193, 195, 197, 210] [13, 26, 36, 99, 171, 179, 182, 194, 209] [27, 36, 37, 67, 176, 184, 195, 216, 223] [27, 28, 49, 101, 180, 182, 184, 196, 220] [29, 49, 50, 68, 176, 178, 181, 197, 198] [10, 30, 40, 95, 180, 191, 194, 198, 222] [0, 31, 41, 78, 175, 189, 199, 203, 204] [2, 23, 32, 97, 187, 193, 194, 196, 200] [7, 33, 52, 80, 177, 201, 203, 205, 217] [21, 34, 44, 106, 177, 186, 190, 202, 216] [35, 44, 45, 74, 172, 183, 187, 192, 203] [5, 35, 36, 107, 176, 188, 190, 192, 204] [5, 37, 53, 76, 183, 185, 189, 205, 206] [0, 17, 38, 103, 181, 188, 199, 202, 206] [3, 39, 48, 86, 182, 197, 207, 211, 212] [7, 31, 40, 104, 195, 201, 202, 204, 208] [8, 14, 41, 88, 173, 184, 209, 211, 213] [1, 29, 42, 109, 172, 184, 194, 198, 210] [1, 43, 51, 82, 178, 191, 195, 200, 211] [11, 43, 44, 110, 183, 196, 198, 200, 212] [11, 45, 55, 84, 191, 193, 197, 213, 214] [3, 25, 46, 108, 189, 196, 207, 210, 214] [4, 9, 47, 94, 169, 190, 205, 215, 218] [15, 22, 48, 96, 179, 192, 216, 218, 219] [6, 37, 49, 111, 178, 192, 202, 206, 217] [6, 50, 54, 90, 185, 199, 203, 208, 218] [18, 19, 51, 92, 170, 199, 201, 205, 219] [10, 16, 52, 102, 174, 198, 213, 220, 221] [12, 13, 53, 98, 168, 193, 207, 211, 221] [26, 27, 54, 100, 175, 207, 209, 213, 222] [20, 21, 55, 105, 171, 201, 215, 218, 223]

Code ID 224-12-21 · download JSON · raw on GitHub