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[[124,10,10]] d ≤
n
124
k
10
d
10
kd²/n
8.065
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, 11, 15, 23, 35, 55, 115, 119, 120, 122]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[38, 41, 43, 69, 77, 103, 107, 115, 119, 123]
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)×124 (2,4)×930 (3,3)×248 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 124 (2,4): 930 (3,3): 248 (3,5): 8556 (3,7): 1240
trapping sets H_Z (1,3)×124 (2,4)×930 (3,3)×248 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 124 (2,4): 930 (3,3): 248 (3,5): 8556 (3,7): 1240

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra code on the cyclic group Z_62 (elements indexed 0..61); supports a = [0, 59, 2] (left regular representation) and b = [0, 20, 12] (right regular representation); H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] (research/kit/group_algebra.py build_2bga).
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-09-20
notes Found by a random support sweep over weight-6 two-block group-algebra codes with n <= 160 (a sweep script kept with the search run (not committed; method in the research note)). Distance is a witness-backed upper bound; surrogate ladder [(200000, 10), (1000000, 10)] trials per side. Advances the weight-6 x unrestricted board; novelty vs the literature unverified.
family generalized 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

[[124,10,10]] weight-6 generalized bicycle code on Z_62

Direction & hypothesis

Track cell unrestricted x weight-6, small n (n <= 160). The board's weight-6 cell is dense below n = 100, but around k = 10 there was a hole between [[90,10,7]] and [[126,12,10]]: nothing with n < 126, k >= 10 and d >= 8. A random sweep over weight-6 (3+3) two-block group-algebra codes, prefiltered on (n, k) against the live board so that only candidates able to land on the frontier were screened, was the cheapest way to probe such holes.

What was searched

Random supports a = {0, g, h}, b = {0, g', h'} (identity fixed by translation symmetry) on cyclic groups Z_N (N = 20..80), tori Z_l x Z_m (20 <= lm <= 80), dihedral groups D_m (m = 10..40) and metacyclic groups Z_n x| Z_k of order 20..80, three quarters at the 3+3 weight split and one quarter at 2+4. Two runs of a sweep script kept with the search run (sweep.py, not committed; the method is described above) (seeds 1 and 2, about 14 minutes total on 2 threads): 401k supports sampled, 36k with even k >= 4 and a reachable nondomination threshold, screened with the kit surrogate (gf2_fast RIS backend) on the ladder 600 -> 5000 -> 20000 trials per side. A candidate advanced a rung only if its witnessed d stayed at or above the smallest d that no board entry with check weight <= 6 dominates.

Evidence trail

Ladder for this code (trials per side -> lightest logical found): 600 -> 10, 5000 -> 10, 20000 -> 10, 200000 -> 10, 1000000 -> 10 (seeds 1..3 and 100, 101). The trusted gate (verify/validate_candidate.py) found no lighter logical in its own RIS refutation pass and labelled the code "advances the weight-6 x unrestricted board". The claim is a witness-backed upper bound d <= 10 on both sides (X witness weight 10, Z witness weight 10); not certified exact.

A second sample of the same parameters, a = {0, 11, 19}, b = {0, 6, 27} on Z_62, gave [[124,10,8]] and is dominated by this code; it was dropped from the staging set.

Dead ends

  • About 91 percent of random weight-6 supports have k < 4 (mostly k = 0 or 2).
  • Metacyclic and dihedral hits with high k ([[144,24,6]] on Z_12 x| Z_6,

    • [[126,18,6]] on Z_7 x| Z_9) had disconnected Tanner graphs (a and b inside a proper subgroup) and are rejected by the verifier; they are copies of smaller codes. The same held for a BB [[112,12,8]] on Z_14 x Z_4 (two copies of [[56,6,8]]) and a BB [[48,16,3]].

  • d = 2 and d = 3 high-rate codes ([[42,28,2]], [[114,52,2]], [[72,24,3]],

    • [[120,40,3]]) pass the nondomination filter but were not staged: they are trivial-distance points, not the target of this direction.

  • No candidate beat a frontier entry outright; every survivor fills a gap.

Tools

Claude Fable 5.1 (Claude Code, unattended workflow). Kit modules research/kit/group_algebra.py, research/kit/surrogate.py (gf2_fast backend, 2 threads), research/kit/submit.py; gate verify/validate_candidate.py. Sweep and packaging scripts: a sweep script kept with the search run (sweep.py, not committed; the method is described above) and package.py. Roughly 15 CPU-minutes for the sweep, 1 minute per finalist for deep refutation.

Reproduction

import sys; sys.path.insert(0, "research/kit") from group_algebra import cyclic_product, build_2bga mul, _ = cyclic_product(62) HX, HZ = build_2bga(mul, [0, 59, 2], [0, 20, 12])

Equivalently a(x) = 1 + x^59 + x^2, b(x) = 1 + x^20 + x^12 in F_2[x]/(x^62 - 1), H_X = [A | B], H_Z = [B^T | A^T].

Parity checks

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