[[278,2,25]] · QEC Challenge

[[278,2,25]] d ≤

278

kd²/n

4.496

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 25, d_Z ≤ 25 · 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 25 · witness weight 25 (claimed upper_bound)

witness operator (support, 25 qubits)

[2, 18, 45, 67, 77, 82, 83, 93, 103, 118, 146, 175, 176, 177, 184, 194, 195, 196, 204, 205, 212, 217, 218, 242, 248]

d_Z 25 · witness weight 25 (claimed upper_bound)

witness operator (support, 25 qubits)

[0, 1, 18, 21, 31, 32, 33, 34, 35, 36, 37, 58, 59, 73, 102, 138, 157, 160, 167, 196, 212, 222, 241, 261, 277]

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)×139 (2,2)×139 (3,2)×139 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 139 (1,4): 139 (2,2): 139 (2,4): 1112 (2,6): 834 (3,2): 139 (3,4): 4726 (3,6): 12510 (3,8): 7506 (3,10): 556

trapping sets H_Z (1,2)×139 (2,2)×139 (3,2)×139 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 139 (1,4): 139 (2,2): 139 (2,4): 1112 (2,6): 834 (3,2): 139 (3,4): 4726 (3,6): 12510 (3,8): 7506 (3,10): 556

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_278_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

[[278,2,25]] — 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 = 4.496.

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_278_w6_X.mtx and codes/GB_278_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 = 139.

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 ≤ 25, 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) = (278,2) 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_278_w6_X.mtx and GB_278_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 139 (max weight 6) · Z-checks 139 (max weight 6)

H_X (139 checks, sparse supports)

[0, 138, 139, 203, 239, 268] [0, 1, 140, 204, 240, 269] [1, 2, 141, 205, 241, 270] [2, 3, 142, 206, 242, 271] [3, 4, 143, 207, 243, 272] [4, 5, 144, 208, 244, 273] [5, 6, 145, 209, 245, 274] [6, 7, 146, 210, 246, 275] [7, 8, 147, 211, 247, 276] [8, 9, 148, 212, 248, 277] [9, 10, 139, 149, 213, 249] [10, 11, 140, 150, 214, 250] [11, 12, 141, 151, 215, 251] [12, 13, 142, 152, 216, 252] [13, 14, 143, 153, 217, 253] [14, 15, 144, 154, 218, 254] [15, 16, 145, 155, 219, 255] [16, 17, 146, 156, 220, 256] [17, 18, 147, 157, 221, 257] [18, 19, 148, 158, 222, 258] [19, 20, 149, 159, 223, 259] [20, 21, 150, 160, 224, 260] [21, 22, 151, 161, 225, 261] [22, 23, 152, 162, 226, 262] [23, 24, 153, 163, 227, 263] [24, 25, 154, 164, 228, 264] [25, 26, 155, 165, 229, 265] [26, 27, 156, 166, 230, 266] [27, 28, 157, 167, 231, 267] [28, 29, 158, 168, 232, 268] [29, 30, 159, 169, 233, 269] [30, 31, 160, 170, 234, 270] [31, 32, 161, 171, 235, 271] [32, 33, 162, 172, 236, 272] [33, 34, 163, 173, 237, 273] [34, 35, 164, 174, 238, 274] [35, 36, 165, 175, 239, 275] [36, 37, 166, 176, 240, 276] [37, 38, 167, 177, 241, 277] [38, 39, 139, 168, 178, 242] [39, 40, 140, 169, 179, 243] [40, 41, 141, 170, 180, 244] [41, 42, 142, 171, 181, 245] [42, 43, 143, 172, 182, 246] [43, 44, 144, 173, 183, 247] [44, 45, 145, 174, 184, 248] [45, 46, 146, 175, 185, 249] [46, 47, 147, 176, 186, 250] [47, 48, 148, 177, 187, 251] [48, 49, 149, 178, 188, 252] [49, 50, 150, 179, 189, 253] [50, 51, 151, 180, 190, 254] [51, 52, 152, 181, 191, 255] [52, 53, 153, 182, 192, 256] [53, 54, 154, 183, 193, 257] [54, 55, 155, 184, 194, 258] [55, 56, 156, 185, 195, 259] [56, 57, 157, 186, 196, 260] [57, 58, 158, 187, 197, 261] [58, 59, 159, 188, 198, 262] [59, 60, 160, 189, 199, 263] [60, 61, 161, 190, 200, 264] [61, 62, 162, 191, 201, 265] [62, 63, 163, 192, 202, 266] [63, 64, 164, 193, 203, 267] [64, 65, 165, 194, 204, 268] [65, 66, 166, 195, 205, 269] [66, 67, 167, 196, 206, 270] [67, 68, 168, 197, 207, 271] [68, 69, 169, 198, 208, 272] [69, 70, 170, 199, 209, 273] [70, 71, 171, 200, 210, 274] [71, 72, 172, 201, 211, 275] [72, 73, 173, 202, 212, 276] [73, 74, 174, 203, 213, 277] [74, 75, 139, 175, 204, 214] [75, 76, 140, 176, 205, 215] [76, 77, 141, 177, 206, 216] [77, 78, 142, 178, 207, 217] [78, 79, 143, 179, 208, 218] [79, 80, 144, 180, 209, 219] [80, 81, 145, 181, 210, 220] [81, 82, 146, 182, 211, 221] [82, 83, 147, 183, 212, 222] [83, 84, 148, 184, 213, 223] [84, 85, 149, 185, 214, 224] [85, 86, 150, 186, 215, 225] [86, 87, 151, 187, 216, 226] [87, 88, 152, 188, 217, 227] [88, 89, 153, 189, 218, 228] [89, 90, 154, 190, 219, 229] [90, 91, 155, 191, 220, 230] [91, 92, 156, 192, 221, 231] [92, 93, 157, 193, 222, 232] [93, 94, 158, 194, 223, 233] [94, 95, 159, 195, 224, 234] [95, 96, 160, 196, 225, 235] [96, 97, 161, 197, 226, 236] [97, 98, 162, 198, 227, 237] [98, 99, 163, 199, 228, 238] [99, 100, 164, 200, 229, 239] [100, 101, 165, 201, 230, 240] [101, 102, 166, 202, 231, 241] [102, 103, 167, 203, 232, 242] [103, 104, 168, 204, 233, 243] [104, 105, 169, 205, 234, 244] [105, 106, 170, 206, 235, 245] [106, 107, 171, 207, 236, 246] [107, 108, 172, 208, 237, 247] [108, 109, 173, 209, 238, 248] [109, 110, 174, 210, 239, 249] [110, 111, 175, 211, 240, 250] [111, 112, 176, 212, 241, 251] [112, 113, 177, 213, 242, 252] [113, 114, 178, 214, 243, 253] [114, 115, 179, 215, 244, 254] [115, 116, 180, 216, 245, 255] [116, 117, 181, 217, 246, 256] [117, 118, 182, 218, 247, 257] [118, 119, 183, 219, 248, 258] [119, 120, 184, 220, 249, 259] [120, 121, 185, 221, 250, 260] [121, 122, 186, 222, 251, 261] [122, 123, 187, 223, 252, 262] [123, 124, 188, 224, 253, 263] [124, 125, 189, 225, 254, 264] [125, 126, 190, 226, 255, 265] [126, 127, 191, 227, 256, 266] [127, 128, 192, 228, 257, 267] [128, 129, 193, 229, 258, 268] [129, 130, 194, 230, 259, 269] [130, 131, 195, 231, 260, 270] [131, 132, 196, 232, 261, 271] [132, 133, 197, 233, 262, 272] [133, 134, 198, 234, 263, 273] [134, 135, 199, 235, 264, 274] [135, 136, 200, 236, 265, 275] [136, 137, 201, 237, 266, 276] [137, 138, 202, 238, 267, 277]

H_Z (139 checks, sparse supports)

[0, 10, 39, 75, 139, 140] [1, 11, 40, 76, 140, 141] [2, 12, 41, 77, 141, 142] [3, 13, 42, 78, 142, 143] [4, 14, 43, 79, 143, 144] [5, 15, 44, 80, 144, 145] [6, 16, 45, 81, 145, 146] [7, 17, 46, 82, 146, 147] [8, 18, 47, 83, 147, 148] [9, 19, 48, 84, 148, 149] [10, 20, 49, 85, 149, 150] [11, 21, 50, 86, 150, 151] [12, 22, 51, 87, 151, 152] [13, 23, 52, 88, 152, 153] [14, 24, 53, 89, 153, 154] [15, 25, 54, 90, 154, 155] [16, 26, 55, 91, 155, 156] [17, 27, 56, 92, 156, 157] [18, 28, 57, 93, 157, 158] [19, 29, 58, 94, 158, 159] [20, 30, 59, 95, 159, 160] [21, 31, 60, 96, 160, 161] [22, 32, 61, 97, 161, 162] [23, 33, 62, 98, 162, 163] [24, 34, 63, 99, 163, 164] [25, 35, 64, 100, 164, 165] [26, 36, 65, 101, 165, 166] [27, 37, 66, 102, 166, 167] [28, 38, 67, 103, 167, 168] [29, 39, 68, 104, 168, 169] [30, 40, 69, 105, 169, 170] [31, 41, 70, 106, 170, 171] [32, 42, 71, 107, 171, 172] [33, 43, 72, 108, 172, 173] [34, 44, 73, 109, 173, 174] [35, 45, 74, 110, 174, 175] [36, 46, 75, 111, 175, 176] [37, 47, 76, 112, 176, 177] [38, 48, 77, 113, 177, 178] [39, 49, 78, 114, 178, 179] [40, 50, 79, 115, 179, 180] [41, 51, 80, 116, 180, 181] [42, 52, 81, 117, 181, 182] [43, 53, 82, 118, 182, 183] [44, 54, 83, 119, 183, 184] [45, 55, 84, 120, 184, 185] [46, 56, 85, 121, 185, 186] [47, 57, 86, 122, 186, 187] [48, 58, 87, 123, 187, 188] [49, 59, 88, 124, 188, 189] [50, 60, 89, 125, 189, 190] [51, 61, 90, 126, 190, 191] [52, 62, 91, 127, 191, 192] [53, 63, 92, 128, 192, 193] [54, 64, 93, 129, 193, 194] [55, 65, 94, 130, 194, 195] [56, 66, 95, 131, 195, 196] [57, 67, 96, 132, 196, 197] [58, 68, 97, 133, 197, 198] [59, 69, 98, 134, 198, 199] [60, 70, 99, 135, 199, 200] [61, 71, 100, 136, 200, 201] [62, 72, 101, 137, 201, 202] [63, 73, 102, 138, 202, 203] [0, 64, 74, 103, 203, 204] [1, 65, 75, 104, 204, 205] [2, 66, 76, 105, 205, 206] [3, 67, 77, 106, 206, 207] [4, 68, 78, 107, 207, 208] [5, 69, 79, 108, 208, 209] [6, 70, 80, 109, 209, 210] [7, 71, 81, 110, 210, 211] [8, 72, 82, 111, 211, 212] [9, 73, 83, 112, 212, 213] [10, 74, 84, 113, 213, 214] [11, 75, 85, 114, 214, 215] [12, 76, 86, 115, 215, 216] [13, 77, 87, 116, 216, 217] [14, 78, 88, 117, 217, 218] [15, 79, 89, 118, 218, 219] [16, 80, 90, 119, 219, 220] [17, 81, 91, 120, 220, 221] [18, 82, 92, 121, 221, 222] [19, 83, 93, 122, 222, 223] [20, 84, 94, 123, 223, 224] [21, 85, 95, 124, 224, 225] [22, 86, 96, 125, 225, 226] [23, 87, 97, 126, 226, 227] [24, 88, 98, 127, 227, 228] [25, 89, 99, 128, 228, 229] [26, 90, 100, 129, 229, 230] [27, 91, 101, 130, 230, 231] [28, 92, 102, 131, 231, 232] [29, 93, 103, 132, 232, 233] [30, 94, 104, 133, 233, 234] [31, 95, 105, 134, 234, 235] [32, 96, 106, 135, 235, 236] [33, 97, 107, 136, 236, 237] [34, 98, 108, 137, 237, 238] [35, 99, 109, 138, 238, 239] [0, 36, 100, 110, 239, 240] [1, 37, 101, 111, 240, 241] [2, 38, 102, 112, 241, 242] [3, 39, 103, 113, 242, 243] [4, 40, 104, 114, 243, 244] [5, 41, 105, 115, 244, 245] [6, 42, 106, 116, 245, 246] [7, 43, 107, 117, 246, 247] [8, 44, 108, 118, 247, 248] [9, 45, 109, 119, 248, 249] [10, 46, 110, 120, 249, 250] [11, 47, 111, 121, 250, 251] [12, 48, 112, 122, 251, 252] [13, 49, 113, 123, 252, 253] [14, 50, 114, 124, 253, 254] [15, 51, 115, 125, 254, 255] [16, 52, 116, 126, 255, 256] [17, 53, 117, 127, 256, 257] [18, 54, 118, 128, 257, 258] [19, 55, 119, 129, 258, 259] [20, 56, 120, 130, 259, 260] [21, 57, 121, 131, 260, 261] [22, 58, 122, 132, 261, 262] [23, 59, 123, 133, 262, 263] [24, 60, 124, 134, 263, 264] [25, 61, 125, 135, 264, 265] [26, 62, 126, 136, 265, 266] [27, 63, 127, 137, 266, 267] [28, 64, 128, 138, 267, 268] [0, 29, 65, 129, 268, 269] [1, 30, 66, 130, 269, 270] [2, 31, 67, 131, 270, 271] [3, 32, 68, 132, 271, 272] [4, 33, 69, 133, 272, 273] [5, 34, 70, 134, 273, 274] [6, 35, 71, 135, 274, 275] [7, 36, 72, 136, 275, 276] [8, 37, 73, 137, 276, 277] [9, 38, 74, 138, 139, 277]

Code ID 278-2-25 · download JSON · raw on GitHub