[[262,2,23]] · QEC Challenge

[[262,2,23]] d ≤

262

kd²/n

4.038

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 23, d_Z ≤ 23 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 23 · witness weight 23 (claimed upper_bound)

witness operator (support, 23 qubits)

[0, 7, 15, 32, 55, 63, 71, 87, 106, 138, 163, 170, 194, 202, 206, 207, 208, 209, 219, 220, 229, 230, 253]

d_Z 23 · witness weight 23 (claimed upper_bound)

witness operator (support, 23 qubits)

[7, 8, 9, 10, 22, 23, 31, 62, 79, 95, 110, 118, 127, 128, 146, 153, 161, 185, 202, 210, 217, 241, 260]

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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×131 (2,2)×131 (3,2)×131 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 131 (1,4): 131 (2,2): 131 (2,4): 1048 (2,6): 786 (3,2): 131 (3,4): 4454 (3,6): 11790 (3,8): 7074 (3,10): 524

trapping sets H_Z (1,2)×131 (2,2)×131 (3,2)×131 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 131 (1,4): 131 (2,2): 131 (2,4): 1048 (2,6): 786 (3,2): 131 (3,4): 4454 (3,6): 11790 (3,8): 7074 (3,10): 524

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko) from github.com/QEC-pages/GB-codes, gb-codes.zip, codes/GB_262_w6_X.mtx and _Z.mtx.

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

notes Literature reproduction of a published generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko), from github.com/QEC-pages/GB-codes gb-codes.zip. Checked against the live board: not equivalent to any existing entry.

family generalized bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[262,2,23]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-6 frontier for hackathon issue #1155. This is one of the original generalized bicycle codes of Renyu Wang and Leonid P. Pryadko (arXiv:2306.16400, "Distance bounds for generalized bicycle codes"). It was absent from the board at the time of submission, and the trusted validator reports that it advances the weight-6 unrestricted cell. Efficiency kd²/n = 4.038.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts. From github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip, the codes/GB_262_w6_X.mtx and codes/GB_262_w6_Z.mtx pair. These are the actual H_X and H_Z parity-check matrices shipped by the authors; no reconstruction from group parameters was needed. The code is a two-block generalized bicycle code over the circulant ring Z_{ell} with ell = 131.

Evidence trail

The trusted candidate gate returned passed: true — structural verification passed (CSS commutation, connectivity, k = 2), and the random-seed refutation found no lighter logical than the claimed distance. Distance is a witness-backed upper bound d ≤ 23, not an exact certificate. The published distance is consistent with this run.

Dead ends

The same repo ships 72 GB codes; most are dominated by existing board entries or already submitted (the 2BGA-codes parameter mining covered a different parameterization). This (n,k) = (262,2) pair survived the board-dominance screen, the within-pool Pareto filter, and the trusted gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; NumPy, the repository submission builder (research/kit/submit.py) and the trusted validator (verify/validate_candidate.py). Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Read the two Matrix Market files GB_262_w6_X.mtx and GB_262_w6_Z.mtx from github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip. Each is a 0/1 coordinate matrix: header line nchecks n nnz, then i j 1 entries (1-based). Build dense H_X, H_Z, verify CSS commutation H_X·H_Zᵀ = 0, and pass both to the repository submission builder with confidence upper_bound. Save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 131 (max weight 6) · Z-checks 131 (max weight 6)

H_X (131 checks, sparse supports)

[0, 130, 131, 230, 245, 253] [0, 1, 132, 231, 246, 254] [1, 2, 133, 232, 247, 255] [2, 3, 134, 233, 248, 256] [3, 4, 135, 234, 249, 257] [4, 5, 136, 235, 250, 258] [5, 6, 137, 236, 251, 259] [6, 7, 138, 237, 252, 260] [7, 8, 139, 238, 253, 261] [8, 9, 131, 140, 239, 254] [9, 10, 132, 141, 240, 255] [10, 11, 133, 142, 241, 256] [11, 12, 134, 143, 242, 257] [12, 13, 135, 144, 243, 258] [13, 14, 136, 145, 244, 259] [14, 15, 137, 146, 245, 260] [15, 16, 138, 147, 246, 261] [16, 17, 131, 139, 148, 247] [17, 18, 132, 140, 149, 248] [18, 19, 133, 141, 150, 249] [19, 20, 134, 142, 151, 250] [20, 21, 135, 143, 152, 251] [21, 22, 136, 144, 153, 252] [22, 23, 137, 145, 154, 253] [23, 24, 138, 146, 155, 254] [24, 25, 139, 147, 156, 255] [25, 26, 140, 148, 157, 256] [26, 27, 141, 149, 158, 257] [27, 28, 142, 150, 159, 258] [28, 29, 143, 151, 160, 259] [29, 30, 144, 152, 161, 260] [30, 31, 145, 153, 162, 261] [31, 32, 131, 146, 154, 163] [32, 33, 132, 147, 155, 164] [33, 34, 133, 148, 156, 165] [34, 35, 134, 149, 157, 166] [35, 36, 135, 150, 158, 167] [36, 37, 136, 151, 159, 168] [37, 38, 137, 152, 160, 169] [38, 39, 138, 153, 161, 170] [39, 40, 139, 154, 162, 171] [40, 41, 140, 155, 163, 172] [41, 42, 141, 156, 164, 173] [42, 43, 142, 157, 165, 174] [43, 44, 143, 158, 166, 175] [44, 45, 144, 159, 167, 176] [45, 46, 145, 160, 168, 177] [46, 47, 146, 161, 169, 178] [47, 48, 147, 162, 170, 179] [48, 49, 148, 163, 171, 180] [49, 50, 149, 164, 172, 181] [50, 51, 150, 165, 173, 182] [51, 52, 151, 166, 174, 183] [52, 53, 152, 167, 175, 184] [53, 54, 153, 168, 176, 185] [54, 55, 154, 169, 177, 186] [55, 56, 155, 170, 178, 187] [56, 57, 156, 171, 179, 188] [57, 58, 157, 172, 180, 189] [58, 59, 158, 173, 181, 190] [59, 60, 159, 174, 182, 191] [60, 61, 160, 175, 183, 192] [61, 62, 161, 176, 184, 193] [62, 63, 162, 177, 185, 194] [63, 64, 163, 178, 186, 195] [64, 65, 164, 179, 187, 196] [65, 66, 165, 180, 188, 197] [66, 67, 166, 181, 189, 198] [67, 68, 167, 182, 190, 199] [68, 69, 168, 183, 191, 200] [69, 70, 169, 184, 192, 201] [70, 71, 170, 185, 193, 202] [71, 72, 171, 186, 194, 203] [72, 73, 172, 187, 195, 204] [73, 74, 173, 188, 196, 205] [74, 75, 174, 189, 197, 206] [75, 76, 175, 190, 198, 207] [76, 77, 176, 191, 199, 208] [77, 78, 177, 192, 200, 209] [78, 79, 178, 193, 201, 210] [79, 80, 179, 194, 202, 211] [80, 81, 180, 195, 203, 212] [81, 82, 181, 196, 204, 213] [82, 83, 182, 197, 205, 214] [83, 84, 183, 198, 206, 215] [84, 85, 184, 199, 207, 216] [85, 86, 185, 200, 208, 217] [86, 87, 186, 201, 209, 218] [87, 88, 187, 202, 210, 219] [88, 89, 188, 203, 211, 220] [89, 90, 189, 204, 212, 221] [90, 91, 190, 205, 213, 222] [91, 92, 191, 206, 214, 223] [92, 93, 192, 207, 215, 224] [93, 94, 193, 208, 216, 225] [94, 95, 194, 209, 217, 226] [95, 96, 195, 210, 218, 227] [96, 97, 196, 211, 219, 228] [97, 98, 197, 212, 220, 229] [98, 99, 198, 213, 221, 230] [99, 100, 199, 214, 222, 231] [100, 101, 200, 215, 223, 232] [101, 102, 201, 216, 224, 233] [102, 103, 202, 217, 225, 234] [103, 104, 203, 218, 226, 235] [104, 105, 204, 219, 227, 236] [105, 106, 205, 220, 228, 237] [106, 107, 206, 221, 229, 238] [107, 108, 207, 222, 230, 239] [108, 109, 208, 223, 231, 240] [109, 110, 209, 224, 232, 241] [110, 111, 210, 225, 233, 242] [111, 112, 211, 226, 234, 243] [112, 113, 212, 227, 235, 244] [113, 114, 213, 228, 236, 245] [114, 115, 214, 229, 237, 246] [115, 116, 215, 230, 238, 247] [116, 117, 216, 231, 239, 248] [117, 118, 217, 232, 240, 249] [118, 119, 218, 233, 241, 250] [119, 120, 219, 234, 242, 251] [120, 121, 220, 235, 243, 252] [121, 122, 221, 236, 244, 253] [122, 123, 222, 237, 245, 254] [123, 124, 223, 238, 246, 255] [124, 125, 224, 239, 247, 256] [125, 126, 225, 240, 248, 257] [126, 127, 226, 241, 249, 258] [127, 128, 227, 242, 250, 259] [128, 129, 228, 243, 251, 260] [129, 130, 229, 244, 252, 261]

H_Z (131 checks, sparse supports)

[0, 9, 17, 32, 131, 132] [1, 10, 18, 33, 132, 133] [2, 11, 19, 34, 133, 134] [3, 12, 20, 35, 134, 135] [4, 13, 21, 36, 135, 136] [5, 14, 22, 37, 136, 137] [6, 15, 23, 38, 137, 138] [7, 16, 24, 39, 138, 139] [8, 17, 25, 40, 139, 140] [9, 18, 26, 41, 140, 141] [10, 19, 27, 42, 141, 142] [11, 20, 28, 43, 142, 143] [12, 21, 29, 44, 143, 144] [13, 22, 30, 45, 144, 145] [14, 23, 31, 46, 145, 146] [15, 24, 32, 47, 146, 147] [16, 25, 33, 48, 147, 148] [17, 26, 34, 49, 148, 149] [18, 27, 35, 50, 149, 150] [19, 28, 36, 51, 150, 151] [20, 29, 37, 52, 151, 152] [21, 30, 38, 53, 152, 153] [22, 31, 39, 54, 153, 154] [23, 32, 40, 55, 154, 155] [24, 33, 41, 56, 155, 156] [25, 34, 42, 57, 156, 157] [26, 35, 43, 58, 157, 158] [27, 36, 44, 59, 158, 159] [28, 37, 45, 60, 159, 160] [29, 38, 46, 61, 160, 161] [30, 39, 47, 62, 161, 162] [31, 40, 48, 63, 162, 163] [32, 41, 49, 64, 163, 164] [33, 42, 50, 65, 164, 165] [34, 43, 51, 66, 165, 166] [35, 44, 52, 67, 166, 167] [36, 45, 53, 68, 167, 168] [37, 46, 54, 69, 168, 169] [38, 47, 55, 70, 169, 170] [39, 48, 56, 71, 170, 171] [40, 49, 57, 72, 171, 172] [41, 50, 58, 73, 172, 173] [42, 51, 59, 74, 173, 174] [43, 52, 60, 75, 174, 175] [44, 53, 61, 76, 175, 176] [45, 54, 62, 77, 176, 177] [46, 55, 63, 78, 177, 178] [47, 56, 64, 79, 178, 179] [48, 57, 65, 80, 179, 180] [49, 58, 66, 81, 180, 181] [50, 59, 67, 82, 181, 182] [51, 60, 68, 83, 182, 183] [52, 61, 69, 84, 183, 184] [53, 62, 70, 85, 184, 185] [54, 63, 71, 86, 185, 186] [55, 64, 72, 87, 186, 187] [56, 65, 73, 88, 187, 188] [57, 66, 74, 89, 188, 189] [58, 67, 75, 90, 189, 190] [59, 68, 76, 91, 190, 191] [60, 69, 77, 92, 191, 192] [61, 70, 78, 93, 192, 193] [62, 71, 79, 94, 193, 194] [63, 72, 80, 95, 194, 195] [64, 73, 81, 96, 195, 196] [65, 74, 82, 97, 196, 197] [66, 75, 83, 98, 197, 198] [67, 76, 84, 99, 198, 199] [68, 77, 85, 100, 199, 200] [69, 78, 86, 101, 200, 201] [70, 79, 87, 102, 201, 202] [71, 80, 88, 103, 202, 203] [72, 81, 89, 104, 203, 204] [73, 82, 90, 105, 204, 205] [74, 83, 91, 106, 205, 206] [75, 84, 92, 107, 206, 207] [76, 85, 93, 108, 207, 208] [77, 86, 94, 109, 208, 209] [78, 87, 95, 110, 209, 210] [79, 88, 96, 111, 210, 211] [80, 89, 97, 112, 211, 212] [81, 90, 98, 113, 212, 213] [82, 91, 99, 114, 213, 214] [83, 92, 100, 115, 214, 215] [84, 93, 101, 116, 215, 216] [85, 94, 102, 117, 216, 217] [86, 95, 103, 118, 217, 218] [87, 96, 104, 119, 218, 219] [88, 97, 105, 120, 219, 220] [89, 98, 106, 121, 220, 221] [90, 99, 107, 122, 221, 222] [91, 100, 108, 123, 222, 223] [92, 101, 109, 124, 223, 224] [93, 102, 110, 125, 224, 225] [94, 103, 111, 126, 225, 226] [95, 104, 112, 127, 226, 227] [96, 105, 113, 128, 227, 228] [97, 106, 114, 129, 228, 229] [98, 107, 115, 130, 229, 230] [0, 99, 108, 116, 230, 231] [1, 100, 109, 117, 231, 232] [2, 101, 110, 118, 232, 233] [3, 102, 111, 119, 233, 234] [4, 103, 112, 120, 234, 235] [5, 104, 113, 121, 235, 236] [6, 105, 114, 122, 236, 237] [7, 106, 115, 123, 237, 238] [8, 107, 116, 124, 238, 239] [9, 108, 117, 125, 239, 240] [10, 109, 118, 126, 240, 241] [11, 110, 119, 127, 241, 242] [12, 111, 120, 128, 242, 243] [13, 112, 121, 129, 243, 244] [14, 113, 122, 130, 244, 245] [0, 15, 114, 123, 245, 246] [1, 16, 115, 124, 246, 247] [2, 17, 116, 125, 247, 248] [3, 18, 117, 126, 248, 249] [4, 19, 118, 127, 249, 250] [5, 20, 119, 128, 250, 251] [6, 21, 120, 129, 251, 252] [7, 22, 121, 130, 252, 253] [0, 8, 23, 122, 253, 254] [1, 9, 24, 123, 254, 255] [2, 10, 25, 124, 255, 256] [3, 11, 26, 125, 256, 257] [4, 12, 27, 126, 257, 258] [5, 13, 28, 127, 258, 259] [6, 14, 29, 128, 259, 260] [7, 15, 30, 129, 260, 261] [8, 16, 31, 130, 131, 261]

Code ID 262-2-23 · download JSON · raw on GitHub