[[124,14,11]] · QEC Challenge

[[124,14,11]] d =

124

kd²/n

13.661

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 11, d_Z = 11 · 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 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[11, 16, 20, 35, 43, 77, 80, 81, 85, 89, 112]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[11, 15, 28, 66, 72, 73, 78, 86, 90, 97, 110]

certificate exact, d = 11 · CryptoMiniSat 5.14.7 SAT

X: no logical < 11 exists; Z: no logical < 11 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 4 · H_Z 4 (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)×124 (2,4)×186 (3,4)×310 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 124 (2,4): 186 (2,6): 1364 (3,4): 310 (3,6): 4836 (3,8): 21700 (3,10): 1550

trapping sets H_Z (1,4)×124 (2,4)×186 (3,4)×310 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 124 (2,4): 186 (2,6): 1364 (3,4): 310 (3,6): 4836 (3,8): 21700 (3,10): 1550

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Univariate bicycle code over R_62 = F2[x]/(x⁶²-1); a(x) = 1 + x + x⁴ + x⁷; b(x) = a(x^(2³)) mod (x⁶²-1) (Frobenius coupling); H_X=[A,B], H_Z=[B^T,A^T]. Reconstruction of arXiv:2605.14173v1 Table I row 1.

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

[[124,14,11]] — univariate bicycle (UB) code, arXiv:2605.14173v1 Table I row 1

Direction & hypothesis

Target cell: weight-8 × unrestricted. The univariate bicycle (UB) family of Rabeti–Mahdavifar (arXiv:2605.14173v1) is a generalized-bicycle subclass with the Frobenius coupling b(x) = a(x)^t in R_n = F2[x]/(x^n − 1), t = 2^ℓ, which halves the search space relative to general GB codes while keeping d_X = d_Z. A pre-screen against the current board showed two Table I rows ([[124,14,11]] and [[178,24,13]]) not dominated by any board entry with max check weight ≤ 8, so both were worth a full reconstruction + validation pass.

What was searched

All nine under-cap rows of the paper's Table I were reconstructed exactly (research/ub_sweep.py, committed). Each row: build H_X=[A,B], H_Z=[B^T,A^T] with b = a(x^(2^ℓ)) mod (x^n−1); assert CSS; recompute k; witness search at 4k trials (n ≤ 300) or 12k trials (n > 300), seed 260514173+row; package via submit.make_submission; validate with validate_candidate(refute=True). Reconstructed k matched the paper on 9/9 rows and the witnessed distance matched the paper's claimed d on 9/9 rows.

Evidence trail

Screening/packaging ladder per row: exact CSS → exact k → witness at the

trial count above → gate refutation (fresh seed).

This code: X and Z witnesses both weight 11 (balanced, as the UB structure

predicts); gate refutation found no lighter logical in 7,460 RIS trials (seed 260514951). Claim: d ≤ 11, upper_bound — not certified exact.

Verdict: passed: true, board_advancing: true, no exact duplicate, no

WL-equivalent duplicate.

Dead ends

Seven of nine Table I rows reconstruct cleanly but are dominated on this board (all validator-passing, staged with verdicts):

| Row | Code | Dominated by | |---|---|---| | 2 | [[146,20,8]] | [[104,30,8]], [[128,21,8]], [[136,38,8]] | | 4 | [[204,36,8]] | [[136,38,8]], [[152,42,8]], [[184,50,10]] | | 5 | [[234,26,14]] | [[210,26,14]] | | 6 | [[252,12,14]] | [[252,12,16]] (same n,k, higher d already on board) | | 7 | [[254,14,14]] | [[254,14,16]] | | 8 | [[372,14,12]] | [[254,14,16]], [[288,16,16]], [[340,16,18]], [[360,16,14]] | | 9 | [[378,12,22]] | [[360,12,24]] |

The paper's w=6 rows lose to the board's strong weight-6 GB records; the [[1022,56,21]] row exceeds the verifier's n ≤ 700 cap and was skipped.

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). Compute ~45 min single core for the full nine-row sweep including refutation gates.

Reproduction

uv run python research/ub_sweep.py (committed at research/ub_sweep.py). This row: n = 62, ℓ = 3, a(x) = 1 + x + x⁴ + x⁷, b(x) = a(x⁸) mod (x⁶²−1) = same support (gcd(62,8)=2, but support preserved here), giving [[124,14,d≤11]].

Parity checks

X-checks 62 (max weight 8) · Z-checks 62 (max weight 8)

H_X (62 checks, sparse supports)

[0, 55, 58, 61, 62, 68, 92, 116] [0, 1, 56, 59, 63, 69, 93, 117] [1, 2, 57, 60, 64, 70, 94, 118] [2, 3, 58, 61, 65, 71, 95, 119] [0, 3, 4, 59, 66, 72, 96, 120] [1, 4, 5, 60, 67, 73, 97, 121] [2, 5, 6, 61, 68, 74, 98, 122] [0, 3, 6, 7, 69, 75, 99, 123] [1, 4, 7, 8, 62, 70, 76, 100] [2, 5, 8, 9, 63, 71, 77, 101] [3, 6, 9, 10, 64, 72, 78, 102] [4, 7, 10, 11, 65, 73, 79, 103] [5, 8, 11, 12, 66, 74, 80, 104] [6, 9, 12, 13, 67, 75, 81, 105] [7, 10, 13, 14, 68, 76, 82, 106] [8, 11, 14, 15, 69, 77, 83, 107] [9, 12, 15, 16, 70, 78, 84, 108] [10, 13, 16, 17, 71, 79, 85, 109] [11, 14, 17, 18, 72, 80, 86, 110] [12, 15, 18, 19, 73, 81, 87, 111] [13, 16, 19, 20, 74, 82, 88, 112] [14, 17, 20, 21, 75, 83, 89, 113] [15, 18, 21, 22, 76, 84, 90, 114] [16, 19, 22, 23, 77, 85, 91, 115] [17, 20, 23, 24, 78, 86, 92, 116] [18, 21, 24, 25, 79, 87, 93, 117] [19, 22, 25, 26, 80, 88, 94, 118] [20, 23, 26, 27, 81, 89, 95, 119] [21, 24, 27, 28, 82, 90, 96, 120] [22, 25, 28, 29, 83, 91, 97, 121] [23, 26, 29, 30, 84, 92, 98, 122] [24, 27, 30, 31, 85, 93, 99, 123] [25, 28, 31, 32, 62, 86, 94, 100] [26, 29, 32, 33, 63, 87, 95, 101] [27, 30, 33, 34, 64, 88, 96, 102] [28, 31, 34, 35, 65, 89, 97, 103] [29, 32, 35, 36, 66, 90, 98, 104] [30, 33, 36, 37, 67, 91, 99, 105] [31, 34, 37, 38, 68, 92, 100, 106] [32, 35, 38, 39, 69, 93, 101, 107] [33, 36, 39, 40, 70, 94, 102, 108] [34, 37, 40, 41, 71, 95, 103, 109] [35, 38, 41, 42, 72, 96, 104, 110] [36, 39, 42, 43, 73, 97, 105, 111] [37, 40, 43, 44, 74, 98, 106, 112] [38, 41, 44, 45, 75, 99, 107, 113] [39, 42, 45, 46, 76, 100, 108, 114] [40, 43, 46, 47, 77, 101, 109, 115] [41, 44, 47, 48, 78, 102, 110, 116] [42, 45, 48, 49, 79, 103, 111, 117] [43, 46, 49, 50, 80, 104, 112, 118] [44, 47, 50, 51, 81, 105, 113, 119] [45, 48, 51, 52, 82, 106, 114, 120] [46, 49, 52, 53, 83, 107, 115, 121] [47, 50, 53, 54, 84, 108, 116, 122] [48, 51, 54, 55, 85, 109, 117, 123] [49, 52, 55, 56, 62, 86, 110, 118] [50, 53, 56, 57, 63, 87, 111, 119] [51, 54, 57, 58, 64, 88, 112, 120] [52, 55, 58, 59, 65, 89, 113, 121] [53, 56, 59, 60, 66, 90, 114, 122] [54, 57, 60, 61, 67, 91, 115, 123]

H_Z (62 checks, sparse supports)

[0, 8, 32, 56, 62, 63, 66, 69] [1, 9, 33, 57, 63, 64, 67, 70] [2, 10, 34, 58, 64, 65, 68, 71] [3, 11, 35, 59, 65, 66, 69, 72] [4, 12, 36, 60, 66, 67, 70, 73] [5, 13, 37, 61, 67, 68, 71, 74] [0, 6, 14, 38, 68, 69, 72, 75] [1, 7, 15, 39, 69, 70, 73, 76] [2, 8, 16, 40, 70, 71, 74, 77] [3, 9, 17, 41, 71, 72, 75, 78] [4, 10, 18, 42, 72, 73, 76, 79] [5, 11, 19, 43, 73, 74, 77, 80] [6, 12, 20, 44, 74, 75, 78, 81] [7, 13, 21, 45, 75, 76, 79, 82] [8, 14, 22, 46, 76, 77, 80, 83] [9, 15, 23, 47, 77, 78, 81, 84] [10, 16, 24, 48, 78, 79, 82, 85] [11, 17, 25, 49, 79, 80, 83, 86] [12, 18, 26, 50, 80, 81, 84, 87] [13, 19, 27, 51, 81, 82, 85, 88] [14, 20, 28, 52, 82, 83, 86, 89] [15, 21, 29, 53, 83, 84, 87, 90] [16, 22, 30, 54, 84, 85, 88, 91] [17, 23, 31, 55, 85, 86, 89, 92] [18, 24, 32, 56, 86, 87, 90, 93] [19, 25, 33, 57, 87, 88, 91, 94] [20, 26, 34, 58, 88, 89, 92, 95] [21, 27, 35, 59, 89, 90, 93, 96] [22, 28, 36, 60, 90, 91, 94, 97] [23, 29, 37, 61, 91, 92, 95, 98] [0, 24, 30, 38, 92, 93, 96, 99] [1, 25, 31, 39, 93, 94, 97, 100] [2, 26, 32, 40, 94, 95, 98, 101] [3, 27, 33, 41, 95, 96, 99, 102] [4, 28, 34, 42, 96, 97, 100, 103] [5, 29, 35, 43, 97, 98, 101, 104] [6, 30, 36, 44, 98, 99, 102, 105] [7, 31, 37, 45, 99, 100, 103, 106] [8, 32, 38, 46, 100, 101, 104, 107] [9, 33, 39, 47, 101, 102, 105, 108] [10, 34, 40, 48, 102, 103, 106, 109] [11, 35, 41, 49, 103, 104, 107, 110] [12, 36, 42, 50, 104, 105, 108, 111] [13, 37, 43, 51, 105, 106, 109, 112] [14, 38, 44, 52, 106, 107, 110, 113] [15, 39, 45, 53, 107, 108, 111, 114] [16, 40, 46, 54, 108, 109, 112, 115] [17, 41, 47, 55, 109, 110, 113, 116] [18, 42, 48, 56, 110, 111, 114, 117] [19, 43, 49, 57, 111, 112, 115, 118] [20, 44, 50, 58, 112, 113, 116, 119] [21, 45, 51, 59, 113, 114, 117, 120] [22, 46, 52, 60, 114, 115, 118, 121] [23, 47, 53, 61, 115, 116, 119, 122] [0, 24, 48, 54, 116, 117, 120, 123] [1, 25, 49, 55, 62, 117, 118, 121] [2, 26, 50, 56, 63, 118, 119, 122] [3, 27, 51, 57, 64, 119, 120, 123] [4, 28, 52, 58, 62, 65, 120, 121] [5, 29, 53, 59, 63, 66, 121, 122] [6, 30, 54, 60, 64, 67, 122, 123] [7, 31, 55, 61, 62, 65, 68, 123]

Code ID 124-14-11 · download JSON · raw on GitHub