[[210,12,10]] · QEC Challenge

[[210,12,10]] d =

210

kd²/n

5.714

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness found by @FarLab · verify/ris_gpu --pair-depth 24 · found at 1.5×10⁷ trials · survived 3×10⁷ trials · 2026-08-19

witness operator (support, 10 qubits)

[106, 113, 127, 134, 148, 155, 169, 176, 190, 197]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness found by @FarLab · verify/ris_gpu --pair-depth 24 · found at 1.5×10⁷ trials · survived 3×10⁷ trials · 2026-08-19

witness operator (support, 10 qubits)

[2, 16, 23, 37, 44, 58, 65, 79, 86, 100]

certificate exact, d = 10 · CryptoMiniSat 5.14.7 SAT

X: no logical < 10 exists; Z: no logical < 10 exists

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 6 · H_Z 6 (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 3 · H_Z 3

trapping sets H_X (1,3)×210 (2,4)×1575 (3,3)×245 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 210 (2,4): 1575 (3,3): 245 (3,5): 15015 (3,7): 2100

trapping sets H_Z (1,3)×210 (2,4)×1575 (3,3)×245 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 210 (2,4): 1575 (3,3): 245 (3,5): 15015 (3,7): 2100

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty new parameter set claimed by submitter

construction Generalized bicycle (two-block) code on Z_105: H_X=[A|B], H_Z=[B^T|A^T], a(x)=x¹⁴+x⁷⁰+x⁸⁴, b(x)=x¹²+x⁴⁸+x⁵⁷; g=gcd(a,b,x¹⁰⁵-1) irreducible of degree 6, k=2*deg(g)=12, row weight 6. Full circulant check sets kept (redundant rows carry the symmetry).

model Claude Fable 5 (claimed, not verified)

builds on arXiv:2203.17216, arXiv:2306.16400

date 2026-08-19

notes Orbit-structured GB search find; the full Tanner automorphism group acts with 4 orbits on the 12 canonical logical classes. Distance evidence: GPU deep-RIS 15M full-kernel pair trials x 2 seeds per side (CPU-verified witnesses) + sqetch 20Mx2 trials/side; nothing below 10 by any engine. Novelty basis: 2026-08-19 lit pass (Wang-Pryadko 2203.17216 = k=2 family; Lin-Pryadko 2306.16400 = n<=100; no zoo/board hits).

family generalized bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 105 (max weight 6) · Z-checks 105 (max weight 6)

H_X (105 checks, sparse supports)

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

H_Z (105 checks, sparse supports)

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

Code ID 210-12-10 · download JSON · raw on GitHub