n
168
Issue #2992 asked for the halved CPM pair-partition codes of arXiv:2609.30069 (Definition 1 with the reversing fold of Proposition 4); its (4,10) family went onto the board on 2026-10-13 as [[90,21,10]], [[95,22,11]], [[140,31,14]], and [[200,43,19]]. This entry comes from the follow-on sweep of 2026-10-12 over the paper's (3,8), (3,10), (4,10), and (4,12) families at the lifts where the stabilizer board had no entry at all: the paper publishes no generators and its data repository (ultra-high-rate-qec/design-principles-data) was empty on 2026-10-11, so the families were searched directly over their exponent arrays. For the (4,12) family n = 6P and k = 2P + 3 for every accepted template, and the distance is whatever the search finds.
The parent is a pair-partition CPM code with J = 4 check blocks and L = 12 qubit blocks of size P = 28. X checks come from an exponent array E: row (j, s) has a 1 at qubit (l, s - E[j][l] mod P) for each l. Z checks come from D[j][l] = -E[j][l xor 1] mod P in the same way. E must satisfy the paper's pair-partition rule: each off-diagonal cell (i < j) carries a perfect matching M on the L block columns, edge-disjoint from the matchings already placed in its row and column, and E[i,u] + E[j,u xor 1] = E[i,v] + E[j,v xor 1] mod P for every matched pair {u, v}. The search draws M at random, solves the resulting homogeneous integer system for E with sympy, and reduces a random integer combination of the kernel basis mod P.
The reversing fold identifies qubit t of block l with qubit -t mod P of block l xor 1. Each parent X check then becomes one stabilizer generator on the 6P = 168 surviving qubits (blocks 0, 2, 4, 6, 8, 10): an X leg on each surviving block's position and a Z leg on each partner position, with coinciding legs merging to Y. Every generator has weight at most 12. The 4P = 112 generators span a group of rank 4P - 3 = 109, hence k = 168 - 109 = 59.
A Perlmutter sweep over the (4,12), P = 28 template on 2026-10-12 (NERSC gen0030, job 59705503): 6 processes for 3.2 hours each, 29,903 accepted template records screened at 1,500 Pauli-weight RIS trials apiece, not deduplicated by E. The eight highest readings in the cell then went through the frame screen (12 random per-qubit symplectic frames x 1,000,000 gf2_fast trials per chunk on the symplectic doubling, jobs 59731365 to 59731368, 2026-10-13 to 14), 120 frames for this one, which came out best of the eight at 12. Staged and gated on 2026-10-14 (job 59777052), it then held 12 through 250 further frames x 1,000,000 trials with fresh seeds (job 59779389, 2026-10-14), where the lightest logical appeared 77 times and nothing lighter appeared; the job was killed by a node failure at about eight of its twelve hours, so the count is what had been written by then. The other seven templates in the cell are parameter twins and are not filed.
[15, 22, 51, 74, 141, 146, 156] (positions in both carry Y), stored in the JSON; it commutes with every generator and lies outside the stabilizer group, as the gate re-checks.
minimizes |x| + |z|, so a logical with Y legs looks heavier than its Pauli weight and can be missed. The frame screen draws a random single-qubit symplectic frame per qubit, runs the doubled search, maps each find back, and scores it by Pauli weight. On a (4,10), P = 40 code of this family, 12 frames x 500,000 trials found Pauli weight 15 where 100,000,000 unframed trials read 19, so unframed non-refutations of these codes are weak evidence, and every count above is framed.
verify/validate_candidate.py through research/kit/coordination.pyagainst main c17845a2, 2026-10-15): passed; advances the weight-9plus x unrestricted stabilizer board; literature novelty UNVERIFIED.
No board entry of any type at n <= 168 and k >= 59 reaches d >= 5; the best is [[128,66,4]]. On the stabilizer board the cell weight-9plus x unrestricted had nothing near this (n, k); the nearest stabilizer entries are [[188,49,6]].
The paper's Table VII lists [[168,59,13]] at this lift, as an exact value; this search did not reach 13, so the entry is 1 short of the published value.
it or its appendices, and the data repository is empty. Where this entry falls short of a published value, whether the gap is in the template search (the paper does not say how its E were chosen) or in our distance estimate (an upper bound) is open.
loosely; the ranking that fed the frame screen came from it, so better templates may have been screened out early.
Template search halved_pp.py and frame screen frame_screen.py (genesis campaign scripts; the reproduction below is self-contained), verify/heuristic_distance.py Pauli RIS, verify/gf2_fast on Perlmutter (NERSC gen0030), verify/validate_candidate.py. Claude Fable 5.1 (Claude Code) working for @vprusso.
import numpy as np
def halved_checks(P, E):
J, L = len(E), len(E[0])
HX = np.zeros((J * P, L * P), dtype=np.int8)
for j in range(J):
for l in range(L):
for s in range(P):
HX[j * P + s, l * P + (s - E[j][l]) % P] = 1
checks = []
for h in HX:
x, z = [], []
for ri, l in enumerate(range(0, L, 2)):
for t in range(P):
if h[l * P + t]:
x.append(ri * P + t)
if h[(l ^ 1) * P + (-t) % P]:
z.append(ri * P + t)
checks.append({"X": x, "Z": z})
return checks
E = [[6, 12, 26, 18, 7, 5, 17, 18, 8, 20, 2, 9], [18, 27, 25, 17, 24, 22, 0, 24, 21, 10, 21, 3], [4, 13, 8, 7, 0, 9, 1, 20, 22, 1, 17, 12], [3, 5, 13, 8, 7, 24, 24, 23, 19, 7, 25, 12]]
S = halved_checks(28, E) # the 4P generators of codes/168-59-12.json, in order