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[[72,14,8]] d =
n
72
k
14
d
8
kd²/n
12.444
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[12, 13, 21, 23, 49, 54, 56, 69]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[1, 2, 4, 20, 37, 40, 41, 59]
certificate exact, d = 8 · CryptoMiniSat 5.14 SAT
X: no logical < 8 exists; Z: no logical < 8 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)×72 (2,4)×72 (3,4)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 72 (2,4): 72 (2,6): 864 (3,4): 216 (3,6): 2832 (3,8): 12024 (3,10): 1224
trapping sets H_Z (1,4)×72 (2,4)×72 (3,4)×216 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 72 (2,4): 72 (2,6): 864 (3,4): 216 (3,6): 2832 (3,8): 12024 (3,10): 1224

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Weight-8 bivariate bicycle on Z_6 x Z_6. A = 1 + x4 y4 + y5 + x y5, B = 1 + x4 y + x y2 + x3 y2. From Lu, Yang, Guo (2026), arXiv:2609.06572, Table 2 entry 1. Exact distance certified by bit-mask DFS.
builds on arXiv:2609.06572
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

[[72,14,8]] — weight-8 bivariate bicycle on Z_6 x Z_6

Direction & hypothesis

Targeted the unrestricted / weight-8 cell at n=72. The existing best weight-8 code at this length was [[72,10,9]] (eff=11.2). Weight-8 BB codes on small grids are underexplored relative to the canonical weight-6 [[72,12,6]] (eff=6), and the algebraic structure theory of arXiv:2609.06572 shows that weight-4 generator polynomials on Z_6 x Z_6 can reach k=14 with exact d=8.

What was searched

Reconstructed the [[72,14,8]] code from the census of arXiv:2609.06572 (Lu, Yang, Guo, Sep 2026). The paper's pipeline sampled 2x10^4 constant-term-normalized pairs (A,B) with wt(A)=wt(B)=4 on the (6,6) grid, screened for k in [4,40] and d>=6, then certified all distances exactly via bit-mask DFS with translation-symmetry pruning. The submitted code is the top entry from their Table 2: A = 1 + x^4 y^4 + y^5 + x y^5, B = 1 + x^4 y + x y^2 + x^3 y^2.

Evidence trail

  • CSS commutation: verified (H_X H_Z^T = 0 over GF(2))
  • k = 14: computed as n - rank(H_X) - rank(H_Z) = 72 - 36 - 22 = 14
  • d_X = 8, d_Z = 8: exact, certified by the paper's bit-mask DFS (no logical

    • operator of weight <= 7 exists); independently confirmed by 20000 RIS trials in the qldpc-challenge verifier (seed 1556915526, no lighter found)

  • Lightest X-logical: weight 8, explicit witness included in submission JSON
  • Lightest Z-logical: weight 8, explicit witness included in submission JSON
  • Dominates at n=72: [[72,8,10]] (k=8 < 14), [[72,8,7]] (k=8, d=7 < 8),

    • [[72,6,6]] (k=6, d=6 < 8), [[72,12,6]] (d=6 < 8)

  • Pareto-optimal vs [[72,10,9]]: higher k (14 vs 10) but lower d (8 vs 9);

    • neither dominates the other

Dead ends

No dead ends for this specific reconstruction — the code was taken directly from a published, exactly-certified census. The coset-based codes from arXiv:2606.17268 ([[96,8,10]], [[112,16,10]]) could not be reconstructed because they require GAP SmallGroup semidirect product groups not available in the pure-NumPy research kit.

Tools

Model: Mimo V2.5. Kit modules used: bb.build_bb (code construction), css.verify_css / css.compute_k (parameter verification), surrogate.distance_rand / surrogate.lightest_logical (witness extraction), submit.make_submission (packaging). Full verification via verify/validate_candidate.py. Approximate compute: <1 minute total.

Reproduction

import sys; sys.path.insert(0, "research/kit")
from bb import build_bb
HX, HZ = build_bb(6, 6,
    A_terms=[(0,0), (4,4), (0,5), (1,5)],
    B_terms=[(0,0), (4,1), (1,2), (3,2)])

Reference: arXiv:2609.06572, Table 2 entry #1 (grid (6,6), weight-8 checks).

Parity checks

X-checks 36 (max weight 8) · Z-checks 36 (max weight 8)
H_X (36 checks, sparse supports)
[0, 5, 11, 28, 36, 44, 56, 61] [0, 1, 6, 29, 37, 45, 57, 62] [1, 2, 7, 24, 38, 46, 58, 63] [2, 3, 8, 25, 39, 47, 59, 64] [3, 4, 9, 26, 40, 42, 54, 65] [4, 5, 10, 27, 41, 43, 55, 60] [6, 11, 17, 34, 42, 50, 62, 67] [6, 7, 12, 35, 43, 51, 63, 68] [7, 8, 13, 30, 44, 52, 64, 69] [8, 9, 14, 31, 45, 53, 65, 70] [9, 10, 15, 32, 46, 48, 60, 71] [10, 11, 16, 33, 47, 49, 61, 66] [4, 12, 17, 23, 37, 48, 56, 68] [5, 12, 13, 18, 38, 49, 57, 69] [0, 13, 14, 19, 39, 50, 58, 70] [1, 14, 15, 20, 40, 51, 59, 71] [2, 15, 16, 21, 41, 52, 54, 66] [3, 16, 17, 22, 36, 53, 55, 67] [10, 18, 23, 29, 38, 43, 54, 62] [11, 18, 19, 24, 39, 44, 55, 63] [6, 19, 20, 25, 40, 45, 56, 64] [7, 20, 21, 26, 41, 46, 57, 65] [8, 21, 22, 27, 36, 47, 58, 60] [9, 22, 23, 28, 37, 42, 59, 61] [16, 24, 29, 35, 44, 49, 60, 68] [17, 24, 25, 30, 45, 50, 61, 69] [12, 25, 26, 31, 46, 51, 62, 70] [13, 26, 27, 32, 47, 52, 63, 71] [14, 27, 28, 33, 42, 53, 64, 66] [15, 28, 29, 34, 43, 48, 65, 67] [5, 22, 30, 35, 38, 50, 55, 66] [0, 23, 30, 31, 39, 51, 56, 67] [1, 18, 31, 32, 40, 52, 57, 68] [2, 19, 32, 33, 41, 53, 58, 69] [3, 20, 33, 34, 36, 48, 59, 70] [4, 21, 34, 35, 37, 49, 54, 71]
H_Z (36 checks, sparse supports)
[0, 17, 22, 34, 36, 37, 50, 67] [1, 12, 23, 35, 37, 38, 51, 68] [2, 13, 18, 30, 38, 39, 52, 69] [3, 14, 19, 31, 39, 40, 53, 70] [4, 15, 20, 32, 40, 41, 48, 71] [5, 16, 21, 33, 36, 41, 49, 66] [4, 6, 23, 28, 37, 42, 43, 56] [5, 7, 18, 29, 38, 43, 44, 57] [0, 8, 19, 24, 39, 44, 45, 58] [1, 9, 20, 25, 40, 45, 46, 59] [2, 10, 21, 26, 41, 46, 47, 54] [3, 11, 22, 27, 36, 42, 47, 55] [10, 12, 29, 34, 43, 48, 49, 62] [11, 13, 24, 35, 44, 49, 50, 63] [6, 14, 25, 30, 45, 50, 51, 64] [7, 15, 26, 31, 46, 51, 52, 65] [8, 16, 27, 32, 47, 52, 53, 60] [9, 17, 28, 33, 42, 48, 53, 61] [4, 16, 18, 35, 49, 54, 55, 68] [5, 17, 19, 30, 50, 55, 56, 69] [0, 12, 20, 31, 51, 56, 57, 70] [1, 13, 21, 32, 52, 57, 58, 71] [2, 14, 22, 33, 53, 58, 59, 66] [3, 15, 23, 34, 48, 54, 59, 67] [5, 10, 22, 24, 38, 55, 60, 61] [0, 11, 23, 25, 39, 56, 61, 62] [1, 6, 18, 26, 40, 57, 62, 63] [2, 7, 19, 27, 41, 58, 63, 64] [3, 8, 20, 28, 36, 59, 64, 65] [4, 9, 21, 29, 37, 54, 60, 65] [11, 16, 28, 30, 44, 61, 66, 67] [6, 17, 29, 31, 45, 62, 67, 68] [7, 12, 24, 32, 46, 63, 68, 69] [8, 13, 25, 33, 47, 64, 69, 70] [9, 14, 26, 34, 42, 65, 70, 71] [10, 15, 27, 35, 43, 60, 66, 71]
Code ID 72-14-8 · download JSON · raw on GitHub