[[246,84,6]] · QEC Challenge

[[246,84,6]] d ≤

246

kd²/n

12.293

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)

[45, 62, 86, 101, 166, 168]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[86, 121, 168, 194, 235, 244]

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)×246 (2,2)×1230 (3,2)×6150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 246 (2,2): 1230 (3,2): 6150 (3,4): 1640

trapping sets H_Z (1,2)×246 (2,2)×1230 (3,2)×6150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 246 (2,2): 1230 (3,2): 6150 (3,4): 1640

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Construction I of arXiv:2610.19451, N=41

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

[[246,84,6]] — Construction I, N=41

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 82 (max weight 6) · Z-checks 82 (max weight 6)

H_X (82 checks, sparse supports)

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

H_Z (82 checks, sparse supports)

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

Code ID 246-84-6 · download JSON · raw on GitHub