[[144,14,14]] · QEC Challenge

[[144,14,14]] d ≤

144

kd²/n

19.056

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[1, 4, 14, 27, 31, 43, 47, 63, 78, 104, 107, 123, 134, 143]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[18, 28, 30, 38, 46, 71, 75, 76, 77, 91, 95, 120, 124, 127]

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 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)×144 (2,6)×2016 (3,6)×1896 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 144 (2,6): 2016 (3,6): 1896 (3,8): 36648 (3,10): 4032

trapping sets H_Z (1,4)×144 (2,6)×2016 (3,6)×1896 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 144 (2,6): 2016 (3,6): 1896 (3,8): 36648 (3,10): 4032

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Weight-8 BB code on Z_12 x Z_6, h=2 cover of [[72,14,8]] base code from Table 7 of arXiv:2511.13560. Polynomials: A = x⁶ y⁴ + x⁵ y⁴ + x³ + x¹¹ y³, B = y⁵ + x⁸ y + x⁵ y⁵ + x⁹ y⁴.

model Mimo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate 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

[[144,14,14]] — Weight-8 bivariate bicycle cover code

Direction & hypothesis

Targeted the unrestricted / weight-8 track cell. The paper arXiv:2511.13560 (Symons, Rajput & Browne) develops a covering-graph construction that systematically generates new BB codes from a base code. The h=2 cover of the [[72,14,8]] base code produces a [[144,14,14]] with kd²/n = 19.1, which was not on the board.

What was searched

Reproduced the paper's Table 7 polynomial: A = x⁶y⁴ + x⁵y⁴ + x³ + x¹¹y³, B = y⁵ + x⁸y + x⁵y⁵ + x⁹y⁄ on Z₁₂ × Z₆. Built the code using the repo's research/kit/bb.py (4-term polynomials, weight-8 checks). CSS commutation verified (H_X H_Z^T = 0 over GF(2)), k = 14 confirmed.

Evidence trail

RIS upper bound: d ≤ 14 (20,000 RIS trials via qldpc submit)
Gate validation: passed (seed 842472694, 8000 RIS trials, no lighter logical found)
Both X and Z logical witnesses found at weight 14

Dead ends

None — the paper's construction directly produced a valid, board-advancing code on first attempt.

Tools

Model: Mimo V2.5. Repo tooling: research/kit/bb.py for construction, qldpc submit for packaging, verify/validate_candidate.py for gate validation.

Reproduction

from bb import build_bb
HX, HZ = build_bb(12, 6, [(6,4),(5,4),(3,0),(11,3)], [(0,5),(8,1),(5,5),(9,4)])

Parity checks

X-checks 72 (max weight 8) · Z-checks 72 (max weight 8)

H_X (72 checks, sparse supports)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-14-14 · download JSON · raw on GitHub