[[28,4,6]] — abelian 2BGA code on Z_7 x Z_2, a = [0, 3, 7, 12], b = [0, 4, 9, 13]

Direction & hypothesis

Track cell: unrestricted x weight-8 (max check weight 8). Target: a point with n' <= 28, k' >= 4, d' >= 6 at check weight <= 8 that no board entry occupies. This code gains on the cell frontier against the entries nearest to it. Nearest board points at n' <= 28: [[20,2,6]] (w=8), [[21,2,6]] (w=8), [[22,2,6]] (w=8), [[24,2,6]] (w=6).

Hypothesis: below n = 110 the board's weight-6 two-block entries are close to exhaustive, but abelian two-block group-algebra (2BGA) codes with larger or unequal supports (|a|, |b| in {3, 4, 5}, check weight 7 to 10) are sampled far more thinly, and every abelian group of a given order (not only cyclic groups and two-factor tori) is a separate family. Two earlier independent seeds over equal 4+4 and 5+5 supports kept producing non-dominated points, so this run used fresh seeds and added the unequal 3+4, 3+5 and 4+5 shapes. Board-relative novelty is from verify/validate_candidate.py; literature novelty is unverified (Lin-Pryadko's 2BGA tables, arXiv:2306.16400, and the bivariate-bicycle literature cover many small abelian groups, and this instance may be in them). Slug note: the board already holds codes/28-4-6.json, a different code with the same [[28,4,6]] but max check weight different (a stabilizer stabilizer code); this one has max check weight 8, so it is filed as 28-4-6-b.

What was searched

H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over F_2[G] with G = Z_7 x Z_2 (|G| = 14) built as group_algebra.cyclic_product(7, 2): elements are the tuples in itertools.product order, index = position in that order (identity = 0). a = [0, 3, 7, 12] and b = [0, 4, 9, 13] are lists of element indices (|a| = 4, |b| = 4, check weight 8). G is abelian, so L(a) and R(b) are commuting multivariate circulants (a bivariate-bicycle-type code when G has two factors).

Sweep (research/kit group_algebra.build_2bga, css.compute_k, surrogate.distance_rand with backend gf2_fast, one thread per candidate, 10-18 worker processes):

• every abelian group of order 14..55 (n = 2|G| <= 110), supports 4+4 (100,000 random pairs per group), 5+5 (30,000 random pairs per group), identity in both a and b, seed 4021: 9,880,000 candidates built, 3,006,860 with k >= 3, each read at 40 gf2_fast RIS trials against the board's best d at that (n, k, weight) (our own open PRs counted as board entries); 2,095 survivors re-read at 400 trials, 752 unique non-dominated records; refinement (at most 3 per distinct (n, k, d, w)) 20,000 trials: 10 of 10 still advancing, 200,000 trials: 10 of 10 still advancing, 2,000,000 trials: 10 of 10 still advancing.
• every abelian group of order 14..55 (n = 2|G| <= 110), supports 3+5 (60,000 random pairs per group), 4+5 (60,000 random pairs per group), 3+4 (60,000 random pairs per group), identity in both a and b, seed 6007: 13,680,000 candidates built, 783,190 with k >= 3, each read at 40 gf2_fast RIS trials against the board's best d at that (n, k, weight) (our own open PRs counted as board entries); 216 survivors re-read at 400 trials, 91 unique non-dominated records; refinement (at most 3 per distinct (n, k, d, w)) 20,000 trials: 15 of 15 still advancing, 200,000 trials: 15 of 15 still advancing, 2,000,000 trials: 15 of 15 still advancing.

Left or right translation of a or b is a qubit relabelling, so fixing the identity in both loses nothing. Every screening reading is an upper bound on d, so the funnel can only discard a candidate whose true distance is also non-advancing.

Evidence trail

Packaging witnesses: surrogate.lightest_logical, 8,000 NumPy trials per side, seed 11 -> X-logical of weight 6, Z-logical of weight 6; d = min = 6.

Deep ladder (fresh seeds, surrogate.distance_rand_witness backend gf2_fast, 8 threads):

| trials per seed | seed | side of lightest | lightest logical | time | |---|---|---|---|---| | 10,000,000 | fast seed 69556 | X | 6 | 6.3 s | | 10,000,000 | fast seed 77475 | X | 6 | 5.8 s | | 10,000,000 | fast seed 85394 | X | 6 | 6.2 s |

Ladder verdict: corroborated (min over rungs = 6, claimed 6). A SAT search for anything lighter (verify/sat_certify.py, 300 s per solve) returned X: status UNSAT; Z: status UNSAT (d_exact flag True). The submission is filed as upper_bound regardless; an exact tier is the server's to award.

Final claim: witness-backed upper bound d <= 6 (confidence: upper_bound), with d_X <= 6 and d_Z <= 6.

Dead ends

• 4+4, 5+5 (seed 4021): the 752 stage-1 records fell on only 4 distinct (n, k, d, w) points not dominated by the board or our open PRs: [[36,4,7]] w8 (x413), [[54,4,10]] w10 (x187), [[28,4,6]] w8 (x151), [[96,16,12]] w10 (x1). Every other draw was dominated already at 40-400 trials, so at this sampling depth these shapes are close to exhausted below n = 110.
• 3+5, 4+5, 3+4 (seed 6007): the 91 stage-1 records fell on only 5 distinct (n, k, d, w) points not dominated by the board or our open PRs: [[54,4,10]] w8 (x46), [[108,4,14]] w7 (x25), [[98,6,14]] w8 (x10), [[90,20,6]] w8 (x7), [[66,4,11]] w9 (x3). Every other draw was dominated already at 40-400 trials, so at this sampling depth these shapes are close to exhausted below n = 110.

Tools

Model: Claude Opus 5.5 (agent harness Claude Code), running on a Brev n2d-highcpu-64 box (gf2_fast via make fast). Kit modules: research/kit/group_algebra.py (build_2bga, cyclic_product), research/kit/css.py, research/kit/surrogate.py, research/kit/prescreen.py, research/kit/submit.py, research/kit/coordination.py (gate). Approximate compute: about 3 CPU-hours for the sweep, refinement and ladder.

Reproduction

import sys; sys.path[:0] = ["research/kit", "verify"]
from group_algebra import build_2bga, cyclic_product
mul, _ = cyclic_product(7, 2)
HX, HZ = build_2bga(mul, [0, 3, 7, 12], [0, 4, 9, 13])   # n = 28, k = 4

Then make_submission(HX, HZ, ..., trials=8000, seed=11) from research/kit/submit.py reproduces the witnesses; the deep ladder used three fresh seeds at the trial counts in the table above.