[[278,2,16]] · QEC Challenge

[[278,2,16]] d ≤

278

kd²/n

1.842

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[7, 32, 60, 89, 118, 142, 143, 144, 145, 146, 170, 171, 199, 228, 256, 257]

d_Z 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[32, 33, 34, 35, 36, 59, 60, 61, 87, 88, 116, 146, 171, 198, 226, 255]

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)×278 (2,2)×834 (3,2)×2502 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 278 (2,2): 834 (3,2): 2502 (3,4): 556

trapping sets H_Z (1,2)×278 (2,2)×834 (3,2)×2502 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 278 (2,2): 834 (3,2): 2502 (3,4): 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_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

[[278,2,16]] — 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 = 1.842.

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_w4_X.mtx and codes/GB_278_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 = 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 ≤ 16, 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_w4_X.mtx and GB_278_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 139 (max weight 4) · Z-checks 139 (max weight 4)

H_X (139 checks, sparse supports)

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

H_Z (139 checks, sparse supports)

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

Code ID 278-2-16 · download JSON · raw on GitHub