[[240,82,6]] · QEC Challenge

[[240,82,6]] d ≤

240

kd²/n

12.3

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)

[33, 73, 107, 176, 210, 221]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[66, 73, 104, 113, 146, 184]

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)×240 (2,2)×1200 (3,2)×6000 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 240 (2,2): 1200 (3,2): 6000 (3,4): 1600

trapping sets H_Z (1,2)×240 (2,2)×1200 (3,2)×6000 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 240 (2,2): 1200 (3,2): 6000 (3,4): 1600

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Construction I of arXiv:2610.19451, N=40

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

[[240,82,6]] — Construction I, N=40

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 80 (max weight 6) · Z-checks 80 (max weight 6)

H_X (80 checks, sparse supports)

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

H_Z (80 checks, sparse supports)

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

Code ID 240-82-6 · download JSON · raw on GitHub