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[[126,12,10]] d ≤
n
126
k
12
d
10
kd²/n
9.524
w
6
X/Z
1

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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 operator (support, 10 qubits)
[1, 7, 14, 21, 42, 48, 56, 69, 89, 124]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[1, 28, 57, 58, 59, 69, 81, 103, 109, 122]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×126 (2,4)×945 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 126 (2,4): 945 (3,3): 252 (3,5): 8694 (3,7): 1260
trapping sets H_Z (1,3)×126 (2,4)×945 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 126 (2,4): 945 (3,3): 252 (3,5): 8694 (3,7): 1260

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Coprime bivariate bicycle on Z_7 x Z_9, pi=xy, a(pi)=1+pi+pi58, b(pi)=1+pi13+pi41. From arXiv:2408.10001v6 (Wang & Mueller, 2026).
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 ≤ 6 (computed)

How this code was found

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

[[126,12,10]] — Coprime bivariate bicycle on Z_7 x Z_9

Direction & hypothesis

Targeted the weight-6 x unrestricted board. The coprime-BB construction from arXiv:2408.10001v6 (Wang & Mueller, Feb 2026) found a [[126,12,10]] code with kd^2/n = 9.52 that dominates 15 existing board entries.

What was searched

Reconstructed from the paper's coprime-BB construction:

  • l=7, m=9 (coprime), pi=xy generates cyclic group of order 63
  • a(pi) = 1 + pi + pi^58, b(pi) = 1 + pi^13 + pi^41
  • Check weight: 6 (3 terms per polynomial)
  • n=126, k=12, d<=10

The coprime-BB construction (Algorithm 2 in the paper) searches over factor polynomials of pi^(lm)+1 over F_2, selecting pairs with GCD = g(pi) to guarantee k = 2*deg(g).

Evidence trail

  • RIS distance surrogate (2000 trials, seed=42): d<=10 (both X and Z sides)
  • CLI witness search (20000 RIS trials): d_X<=10, d_Z<=10
  • Witnesses confirmed by verifier's criteria

Dead ends

  • The paper's [[108,12,6]] was rejected as disconnected (Tanner graph has

    • multiple components).

  • The paper's [[126,8,10]] was dominated by existing entries [[90,8,10]],

    • [[108,8,10]], and [[120,8,12]].

Tools

  • Source: arXiv:2408.10001v6, reconstructed using CRT-based coprime-BB builder
  • Distance: RIS surrogate via research/kit/surrogate.py
  • Packaging: research/kit/submit.py
  • Verification: verify/validate_candidate.py

Reproduction

import sys; sys.path.insert(0, "research/kit"); sys.path.insert(0, "verify")
from math import gcd
assert gcd(7, 9) == 1  # coprime requirement
# pi=xy, a(pi) = 1+pi+pi^58, b(pi) = 1+pi^13+pi^41
# Build via coprime-BB matrix construction on Z_7 x Z_9

Parity checks

X-checks 63 (max weight 6) · Z-checks 63 (max weight 6)
H_X (63 checks, sparse supports)
[0, 1, 58, 63, 76, 104] [1, 2, 59, 64, 77, 105] [2, 3, 60, 65, 78, 106] [3, 4, 61, 66, 79, 107] [4, 5, 62, 67, 80, 108] [0, 5, 6, 68, 81, 109] [1, 6, 7, 69, 82, 110] [2, 7, 8, 70, 83, 111] [3, 8, 9, 71, 84, 112] [4, 9, 10, 72, 85, 113] [5, 10, 11, 73, 86, 114] [6, 11, 12, 74, 87, 115] [7, 12, 13, 75, 88, 116] [8, 13, 14, 76, 89, 117] [9, 14, 15, 77, 90, 118] [10, 15, 16, 78, 91, 119] [11, 16, 17, 79, 92, 120] [12, 17, 18, 80, 93, 121] [13, 18, 19, 81, 94, 122] [14, 19, 20, 82, 95, 123] [15, 20, 21, 83, 96, 124] [16, 21, 22, 84, 97, 125] [17, 22, 23, 63, 85, 98] [18, 23, 24, 64, 86, 99] [19, 24, 25, 65, 87, 100] [20, 25, 26, 66, 88, 101] [21, 26, 27, 67, 89, 102] [22, 27, 28, 68, 90, 103] [23, 28, 29, 69, 91, 104] [24, 29, 30, 70, 92, 105] [25, 30, 31, 71, 93, 106] [26, 31, 32, 72, 94, 107] [27, 32, 33, 73, 95, 108] [28, 33, 34, 74, 96, 109] [29, 34, 35, 75, 97, 110] [30, 35, 36, 76, 98, 111] [31, 36, 37, 77, 99, 112] [32, 37, 38, 78, 100, 113] [33, 38, 39, 79, 101, 114] [34, 39, 40, 80, 102, 115] [35, 40, 41, 81, 103, 116] [36, 41, 42, 82, 104, 117] [37, 42, 43, 83, 105, 118] [38, 43, 44, 84, 106, 119] [39, 44, 45, 85, 107, 120] [40, 45, 46, 86, 108, 121] [41, 46, 47, 87, 109, 122] [42, 47, 48, 88, 110, 123] [43, 48, 49, 89, 111, 124] [44, 49, 50, 90, 112, 125] [45, 50, 51, 63, 91, 113] [46, 51, 52, 64, 92, 114] [47, 52, 53, 65, 93, 115] [48, 53, 54, 66, 94, 116] [49, 54, 55, 67, 95, 117] [50, 55, 56, 68, 96, 118] [51, 56, 57, 69, 97, 119] [52, 57, 58, 70, 98, 120] [53, 58, 59, 71, 99, 121] [54, 59, 60, 72, 100, 122] [55, 60, 61, 73, 101, 123] [56, 61, 62, 74, 102, 124] [0, 57, 62, 75, 103, 125]
H_Z (63 checks, sparse supports)
[0, 22, 50, 63, 68, 125] [1, 23, 51, 63, 64, 69] [2, 24, 52, 64, 65, 70] [3, 25, 53, 65, 66, 71] [4, 26, 54, 66, 67, 72] [5, 27, 55, 67, 68, 73] [6, 28, 56, 68, 69, 74] [7, 29, 57, 69, 70, 75] [8, 30, 58, 70, 71, 76] [9, 31, 59, 71, 72, 77] [10, 32, 60, 72, 73, 78] [11, 33, 61, 73, 74, 79] [12, 34, 62, 74, 75, 80] [0, 13, 35, 75, 76, 81] [1, 14, 36, 76, 77, 82] [2, 15, 37, 77, 78, 83] [3, 16, 38, 78, 79, 84] [4, 17, 39, 79, 80, 85] [5, 18, 40, 80, 81, 86] [6, 19, 41, 81, 82, 87] [7, 20, 42, 82, 83, 88] [8, 21, 43, 83, 84, 89] [9, 22, 44, 84, 85, 90] [10, 23, 45, 85, 86, 91] [11, 24, 46, 86, 87, 92] [12, 25, 47, 87, 88, 93] [13, 26, 48, 88, 89, 94] [14, 27, 49, 89, 90, 95] [15, 28, 50, 90, 91, 96] [16, 29, 51, 91, 92, 97] [17, 30, 52, 92, 93, 98] [18, 31, 53, 93, 94, 99] [19, 32, 54, 94, 95, 100] [20, 33, 55, 95, 96, 101] [21, 34, 56, 96, 97, 102] [22, 35, 57, 97, 98, 103] [23, 36, 58, 98, 99, 104] [24, 37, 59, 99, 100, 105] [25, 38, 60, 100, 101, 106] [26, 39, 61, 101, 102, 107] [27, 40, 62, 102, 103, 108] [0, 28, 41, 103, 104, 109] [1, 29, 42, 104, 105, 110] [2, 30, 43, 105, 106, 111] [3, 31, 44, 106, 107, 112] [4, 32, 45, 107, 108, 113] [5, 33, 46, 108, 109, 114] [6, 34, 47, 109, 110, 115] [7, 35, 48, 110, 111, 116] [8, 36, 49, 111, 112, 117] [9, 37, 50, 112, 113, 118] [10, 38, 51, 113, 114, 119] [11, 39, 52, 114, 115, 120] [12, 40, 53, 115, 116, 121] [13, 41, 54, 116, 117, 122] [14, 42, 55, 117, 118, 123] [15, 43, 56, 118, 119, 124] [16, 44, 57, 119, 120, 125] [17, 45, 58, 63, 120, 121] [18, 46, 59, 64, 121, 122] [19, 47, 60, 65, 122, 123] [20, 48, 61, 66, 123, 124] [21, 49, 62, 67, 124, 125]
Code ID 126-12-10 · download JSON · raw on GitHub