[[282,96,6]] · QEC Challenge

[[282,96,6]] d ≤

282

kd²/n

12.255

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[7, 59, 104, 146, 193, 240]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[14, 43, 72, 202, 260, 278]

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 12 · H_Z 12 (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 · H_Z 2

trapping sets H_X (1,2)×282 (2,2)×1410 (3,2)×7050 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 282 (2,2): 1410 (3,2): 7050 (3,4): 1880

trapping sets H_Z (1,2)×282 (2,2)×1410 (3,2)×7050 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 282 (2,2): 1410 (3,2): 7050 (3,4): 1880

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Construction I of arXiv:2610.19451, N=47

model Beam 1.0 (claimed, not verified)

date 2026-10-17

family other (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

[[282,96,6]] — Construction I, N=47

Direction & hypothesis

Aimed at the CSS weight-6 / unrestricted cell using the constructions of arXiv:2610.19451 (Nourozi, Smarandache, Mitchell). Construction I is a 2x6 circulant family with analytic distance 6 and girth 12; Construction II is a mixed circulant-affine 2-block family with larger certified distances.

What was searched

The paper's fixed constructions were rebuilt from the construction described in arXiv:2610.19451 and screened against the board Pareto frontier. The undominated members listed above were submitted.

Evidence trail

Construction I carries the paper's analytic witnesses (Eq. (12) and its isomorphism mirror), so d <= 6 is a witnessed upper bound. Construction II (q=6) carries a witness found by a fast-RIS ladder; the paper certifies d = 20 by bounded enumeration but does not publish the operators.

Dead ends

Construction I with N = 63 and Construction II with q = 2, 3, 4 pass validation but are dominated by existing board entries and were not submitted.

Tools

Model: Beam 1.0. Repo tooling: verify/qldpc_verify.py.

Reproduction

Rebuild from arXiv:2610.19451: Construction I uses fixed circulant lists a=(0,1,3,8,12,18), b=(0,-5,-5,0,12,12), c=(0,0,2,8,2,8) mod N; Construction II uses the fixed polynomials A(z), B(z) over Z_24 with outer lift q=6.

Parity checks

X-checks 94 (max weight 6) · Z-checks 94 (max weight 6)

H_X (94 checks, sparse supports)

[0, 47, 94, 141, 188, 235] [1, 48, 95, 142, 189, 236] [2, 49, 96, 143, 190, 237] [3, 50, 97, 144, 191, 238] [4, 51, 98, 145, 192, 239] [5, 52, 99, 146, 193, 240] [6, 53, 100, 147, 194, 241] [7, 54, 101, 148, 195, 242] [8, 55, 102, 149, 196, 243] [9, 56, 103, 150, 197, 244] [10, 57, 104, 151, 198, 245] [11, 58, 105, 152, 199, 246] [12, 59, 106, 153, 200, 247] [13, 60, 107, 154, 201, 248] [14, 61, 108, 155, 202, 249] [15, 62, 109, 156, 203, 250] [16, 63, 110, 157, 204, 251] [17, 64, 111, 158, 205, 252] [18, 65, 112, 159, 206, 253] [19, 66, 113, 160, 207, 254] [20, 67, 114, 161, 208, 255] [21, 68, 115, 162, 209, 256] [22, 69, 116, 163, 210, 257] [23, 70, 117, 164, 211, 258] [24, 71, 118, 165, 212, 259] [25, 72, 119, 166, 213, 260] [26, 73, 120, 167, 214, 261] [27, 74, 121, 168, 215, 262] [28, 75, 122, 169, 216, 263] [29, 76, 123, 170, 217, 264] [30, 77, 124, 171, 218, 265] [31, 78, 125, 172, 219, 266] [32, 79, 126, 173, 220, 267] [33, 80, 127, 174, 221, 268] [34, 81, 128, 175, 222, 269] [35, 82, 129, 176, 223, 270] [36, 83, 130, 177, 224, 271] [37, 84, 131, 178, 225, 272] [38, 85, 132, 179, 226, 273] [39, 86, 133, 180, 227, 274] [40, 87, 134, 181, 228, 275] [41, 88, 135, 182, 229, 276] [42, 89, 136, 183, 230, 277] [43, 90, 137, 184, 231, 278] [44, 91, 138, 185, 232, 279] [45, 92, 139, 186, 233, 280] [46, 93, 140, 187, 234, 281] [0, 93, 138, 180, 223, 264] [1, 47, 139, 181, 224, 265] [2, 48, 140, 182, 225, 266] [3, 49, 94, 183, 226, 267] [4, 50, 95, 184, 227, 268] [5, 51, 96, 185, 228, 269] [6, 52, 97, 186, 229, 270] [7, 53, 98, 187, 230, 271] [8, 54, 99, 141, 231, 272] [9, 55, 100, 142, 232, 273] [10, 56, 101, 143, 233, 274] [11, 57, 102, 144, 234, 275] [12, 58, 103, 145, 188, 276] [13, 59, 104, 146, 189, 277] [14, 60, 105, 147, 190, 278] [15, 61, 106, 148, 191, 279] [16, 62, 107, 149, 192, 280] [17, 63, 108, 150, 193, 281] [18, 64, 109, 151, 194, 235] [19, 65, 110, 152, 195, 236] [20, 66, 111, 153, 196, 237] [21, 67, 112, 154, 197, 238] [22, 68, 113, 155, 198, 239] [23, 69, 114, 156, 199, 240] [24, 70, 115, 157, 200, 241] [25, 71, 116, 158, 201, 242] [26, 72, 117, 159, 202, 243] [27, 73, 118, 160, 203, 244] [28, 74, 119, 161, 204, 245] [29, 75, 120, 162, 205, 246] [30, 76, 121, 163, 206, 247] [31, 77, 122, 164, 207, 248] [32, 78, 123, 165, 208, 249] [33, 79, 124, 166, 209, 250] [34, 80, 125, 167, 210, 251] [35, 81, 126, 168, 211, 252] [36, 82, 127, 169, 212, 253] [37, 83, 128, 170, 213, 254] [38, 84, 129, 171, 214, 255] [39, 85, 130, 172, 215, 256] [40, 86, 131, 173, 216, 257] [41, 87, 132, 174, 217, 258] [42, 88, 133, 175, 218, 259] [43, 89, 134, 176, 219, 260] [44, 90, 135, 177, 220, 261] [45, 91, 136, 178, 221, 262] [46, 92, 137, 179, 222, 263]

H_Z (94 checks, sparse supports)

[0, 52, 99, 141, 223, 270] [1, 53, 100, 142, 224, 271] [2, 54, 101, 143, 225, 272] [3, 55, 102, 144, 226, 273] [4, 56, 103, 145, 227, 274] [5, 57, 104, 146, 228, 275] [6, 58, 105, 147, 229, 276] [7, 59, 106, 148, 230, 277] [8, 60, 107, 149, 231, 278] [9, 61, 108, 150, 232, 279] [10, 62, 109, 151, 233, 280] [11, 63, 110, 152, 234, 281] [12, 64, 111, 153, 188, 235] [13, 65, 112, 154, 189, 236] [14, 66, 113, 155, 190, 237] [15, 67, 114, 156, 191, 238] [16, 68, 115, 157, 192, 239] [17, 69, 116, 158, 193, 240] [18, 70, 117, 159, 194, 241] [19, 71, 118, 160, 195, 242] [20, 72, 119, 161, 196, 243] [21, 73, 120, 162, 197, 244] [22, 74, 121, 163, 198, 245] [23, 75, 122, 164, 199, 246] [24, 76, 123, 165, 200, 247] [25, 77, 124, 166, 201, 248] [26, 78, 125, 167, 202, 249] [27, 79, 126, 168, 203, 250] [28, 80, 127, 169, 204, 251] [29, 81, 128, 170, 205, 252] [30, 82, 129, 171, 206, 253] [31, 83, 130, 172, 207, 254] [32, 84, 131, 173, 208, 255] [33, 85, 132, 174, 209, 256] [34, 86, 133, 175, 210, 257] [35, 87, 134, 176, 211, 258] [36, 88, 135, 177, 212, 259] [37, 89, 136, 178, 213, 260] [38, 90, 137, 179, 214, 261] [39, 91, 138, 180, 215, 262] [40, 92, 139, 181, 216, 263] [41, 93, 140, 182, 217, 264] [42, 47, 94, 183, 218, 265] [43, 48, 95, 184, 219, 266] [44, 49, 96, 185, 220, 267] [45, 50, 97, 186, 221, 268] [46, 51, 98, 187, 222, 269] [0, 47, 139, 180, 233, 274] [1, 48, 140, 181, 234, 275] [2, 49, 94, 182, 188, 276] [3, 50, 95, 183, 189, 277] [4, 51, 96, 184, 190, 278] [5, 52, 97, 185, 191, 279] [6, 53, 98, 186, 192, 280] [7, 54, 99, 187, 193, 281] [8, 55, 100, 141, 194, 235] [9, 56, 101, 142, 195, 236] [10, 57, 102, 143, 196, 237] [11, 58, 103, 144, 197, 238] [12, 59, 104, 145, 198, 239] [13, 60, 105, 146, 199, 240] [14, 61, 106, 147, 200, 241] [15, 62, 107, 148, 201, 242] [16, 63, 108, 149, 202, 243] [17, 64, 109, 150, 203, 244] [18, 65, 110, 151, 204, 245] [19, 66, 111, 152, 205, 246] [20, 67, 112, 153, 206, 247] [21, 68, 113, 154, 207, 248] [22, 69, 114, 155, 208, 249] [23, 70, 115, 156, 209, 250] [24, 71, 116, 157, 210, 251] [25, 72, 117, 158, 211, 252] [26, 73, 118, 159, 212, 253] [27, 74, 119, 160, 213, 254] [28, 75, 120, 161, 214, 255] [29, 76, 121, 162, 215, 256] [30, 77, 122, 163, 216, 257] [31, 78, 123, 164, 217, 258] [32, 79, 124, 165, 218, 259] [33, 80, 125, 166, 219, 260] [34, 81, 126, 167, 220, 261] [35, 82, 127, 168, 221, 262] [36, 83, 128, 169, 222, 263] [37, 84, 129, 170, 223, 264] [38, 85, 130, 171, 224, 265] [39, 86, 131, 172, 225, 266] [40, 87, 132, 173, 226, 267] [41, 88, 133, 174, 227, 268] [42, 89, 134, 175, 228, 269] [43, 90, 135, 176, 229, 270] [44, 91, 136, 177, 230, 271] [45, 92, 137, 178, 231, 272] [46, 93, 138, 179, 232, 273]

Code ID 282-96-6 · download JSON · raw on GitHub