[[258,88,6]] · QEC Challenge

[[258,88,6]] d ≤

258

kd²/n

12.279

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)

[73, 114, 115, 116, 152, 153]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[114, 120, 191, 206, 234, 243]

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)×258 (2,2)×1290 (3,2)×6450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 258 (2,2): 1290 (3,2): 6450 (3,4): 1720

trapping sets H_Z (1,2)×258 (2,2)×1290 (3,2)×6450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 258 (2,2): 1290 (3,2): 6450 (3,4): 1720

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Construction I of arXiv:2610.19451, N=43

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

[[258,88,6]] — Construction I, N=43

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 86 (max weight 6) · Z-checks 86 (max weight 6)

H_X (86 checks, sparse supports)

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

H_Z (86 checks, sparse supports)

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

Code ID 258-88-6 · download JSON · raw on GitHub