[[178,24,13]] · QEC Challenge

[[178,24,13]] d =

178

kd²/n

22.787

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 13, d_Z = 13 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[3, 21, 46, 49, 53, 56, 71, 85, 136, 138, 139, 142, 145]

d_Z 13 · witness weight 13 (claimed upper_bound)

witness operator (support, 13 qubits)

[22, 25, 28, 29, 31, 96, 111, 114, 118, 121, 146, 164, 171]

certificate exact, d = 13 · CryptoMiniSat 5.14.7 SAT

X: no logical < 13 exists; Z: no logical < 13 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×178 (2,6)×2492 (3,6)×1602 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 178 (2,6): 2492 (3,6): 1602 (3,8): 47526 (3,10): 4984

trapping sets H_Z (1,4)×178 (2,6)×2492 (3,6)×1602 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 178 (2,6): 2492 (3,6): 1602 (3,8): 47526 (3,10): 4984

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Univariate bicycle code over R_89 = F2[x]/(x⁸⁹-1); a(x) = 1 + x⁹ + x¹⁰ + x¹²; b(x) = a(x^(2⁵)) mod (x⁸⁹-1) (Frobenius coupling, gcd(89,32)=1 so wt(b)=wt(a)); H_X=[A,B], H_Z=[B^T,A^T]. Reconstruction of arXiv:2605.14173v1 Table I row 3.

model Ox Alpha 1.0 (claimed, not verified)

date 2026-08-23

family generalized bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[178,24,13]] — univariate bicycle (UB) code, arXiv:2605.14173v1 Table I row 3

Direction & hypothesis

Target cell: weight-8 × unrestricted. Same campaign as [[124,14,11]] (see its note): reconstruct all under-cap rows of the univariate-bicycle Table I (arXiv:2605.14173v1) and let the trusted validator decide board advancement. This row pre-screened as non-dominated at w ≤ 8 and delivered.

What was searched

Full nine-row sweep (research/ub_sweep.py, committed): for each row build H_X=[A,B], H_Z=[B^T,A^T] with b(x) = a(x^(2^ℓ)) mod (x^n−1); exact CSS and k checks; witness search at 12k trials (this row's size band), seed 260514176; package via submit.make_submission; validate_candidate(refute=True). Reconstructed k = 24 matched the paper; witnessed d = 13 matched the paper's claimed d.

Evidence trail

X and Z witnesses both weight 13 (balanced sides).
Gate refutation: no lighter logical in 8,000 RIS trials (seed 260514953).
Claim: d ≤ 13, upper_bound — not certified exact.
Verdict: passed: true, board_advancing: true, no exact or WL-equivalent

duplicate. Efficiency kd²/n ≈ 23.9 in its cell.

Dead ends

See the [[124,14,11]] note for the seven dominated sibling rows. The pattern: the UB family's w=6 rows are beaten by this board's existing weight-6 GB records, and its high-rate w=8 rows lose to the designed-divisor GB points; the wins are exactly the two mid-size balanced rows.

Tools

Model: Ox Alpha 1.0 (Zed agent). Repo tooling: research/kit/css.py, research/kit/submit.py, research/kit/surrogate.py, verify/validate_candidate.py (untouched). ~45 min single core for the full sweep including refutation gates.

Reproduction

uv run python research/ub_sweep.py (committed at research/ub_sweep.py). This row: n = 89, ℓ = 5, a(x) = 1 + x⁹ + x¹⁰ + x¹², b(x) = a(x³²) mod (x⁸⁹−1); since gcd(89,32)=1 the Frobenius map is a permutation and wt(b)=4, so both check blocks have row weight 8, giving [[178,24,d≤13]].

Parity checks

X-checks 89 (max weight 8) · Z-checks 89 (max weight 8)

H_X (89 checks, sparse supports)

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

H_Z (89 checks, sparse supports)

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

Code ID 178-24-13 · download JSON · raw on GitHub