[[262,2,15]] · QEC Challenge

[[262,2,15]] d ≤

262

kd²/n

1.718

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 107, 117, 127, 218, 228]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[72, 73, 134, 143, 154, 163, 174, 183, 194, 205, 214, 225, 234, 245, 254]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×262 (2,2)×786 (3,2)×2358 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 262 (2,2): 786 (3,2): 2358 (3,4): 524

trapping sets H_Z (1,2)×262 (2,2)×786 (3,2)×2358 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 262 (2,2): 786 (3,2): 2358 (3,4): 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_w4_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 ≤ 4 (computed)

How this code was found

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

[[262,2,15]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-4 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-4 unrestricted cell. Efficiency kd²/n = 1.718.

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_w4_X.mtx and codes/GB_262_w4_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 ≤ 15, 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_w4_X.mtx and GB_262_w4_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 4) · Z-checks 131 (max weight 4)

H_X (131 checks, sparse supports)

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

H_Z (131 checks, sparse supports)

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

Code ID 262-2-15 · download JSON · raw on GitHub