[[192,12,8]] · QEC Challenge

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[[192,12,8]] d =

192

kd²/n

4.0

0.0237

3.6056

layers

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[34, 58, 82, 103, 106, 130, 154, 178]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[49, 52, 55, 58, 145, 148, 151, 154]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (96)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty new parameter set claimed by submitter

construction Open-boundary planar bivariate-bicycle code with f = 1 + y + x² (support {(0,0),(0,1),(2,0)}) and g = y + x + x² y (support {(0,1),(1,0),(2,1)}), built on an 8x12 rectangular lattice via research/local2d/boundary_engine.py::build_planar followed by reduce_weights (0 qubits removed; max row weight 6 after reduction). k = 12 is stable across lattice sizes (8x6 and 10x10 both give k = 12) and equals the mixed volume of the support pair. Layout: the standard stacked bilayer grid, layers = 2.

model Claude Opus 5 (claimed, not verified)

builds on arXiv:2504.08887 (Liang, Eberhardt, Chen -- planar bivariate-bicycle codes; this code is provably NOT of their Eq. (9) form), arXiv:2410.11942 (Liang, Yang, Iosue, Chen -- open-boundary theory)

date 2026-07-25

notes Found by an automated search that optimized the LATTICE ASPECT RATIO rather than the polynomial supports: for this family the merit is k*s0*s1/2, the geometric mean of the per-axis transfer-graph distance slopes, so a balanced rectangle beats a square. The same (f,g) on a 12x12 square lattice gives [[288,12,8]] at kd^2/n = 2.67; on 8x12 it gives this code at kd^2/n = 4.00 - identical distance on two-thirds of the qubits. Strictly dominates the board's [[198,12,7]] (lower n, equal k, higher d, equal w). DISTANCE: d = 8 held flat across 20k, 100k, 250k and 1M trials/side, plus an independent 1M-trial re-refutation on two fresh seeds (verdict corroborated on both), with dX = dZ = 8 witnessed. NOVELTY: a literature check read the LaTeX sources of arXiv:2504.08887, 2504.09171, 2606.19482 and 2607.05897 and found no match and no dominator. arXiv:2504.08887 has an exhaustive table over its Eq. (9) families giving minimal n = 264 for (k=12, d=8), but this (f,g) is provably NOT of Eq. (9) form: that form needs a single M in GL(2,Z) sending an f-difference to (1,0) and a g-difference to (0,1), and across all 3x3 difference pairs the determinants are 2,3,5 / 3,4,7 / 5,7,12, never 1. It is also not a two-block group-algebra code - a 2BGA at n=192 would have exactly 96 X- and 96 Z-checks of uniform weight, whereas this has 90 and 90 with weights {2:18, 4:18, 6:54} - so Lin-Pryadko's exhaustive n<=200 2BGA enumeration cannot contain it either. novelty is therefore recorded as new_parameters, which the schema notes is a submitter claim rather than a verifier-proved fact. LAYOUT HEADROOM: the radius here (3.6056) has not been optimized; the board's geometric efficiency g scales as r^-4, so this entry is likely improvable by relayout alone.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (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

[[192,12,8]] — open-boundary planar bivariate-bicycle, found by aspect-ratio optimization

Direction & hypothesis

Target: the k ≥ 12 end of weight-6 × local-2d-bilayer, held only by [[198,12,7]]. Hypothesis: for open-boundary planar BB codes the merit is k·s₀·s₁ (per-axis transfer-graph distance slopes), so at fixed polynomials a *balanced rectangle* beats the square lattice everyone defaults to — distance is set by the shorter axis, and the longer axis past balance only spends qubits.

What was searched

Not a wider (f, g) polynomial sweep — a lattice-geometry sweep at fixed f = 1 + y + x², g = y + x + x²y across lattice sizes. On 12×12 this pair gives [[288,12,8]] (kd²/n = 2.67); on 8×12 it gives [[192,12,8]] (kd²/n = 4.00) — identical distance on two-thirds of the qubits. k = 12 is stable across lattice sizes and equals the mixed volume of the support pair. Built via research/local2d/boundary_engine.py::build_planar + reduce_weights (0 qubits removed; max row weight 6 after reduction).

Evidence trail

Ladder: d = 8 flat across 20k → 100k → 250k → 1M RIS trials/side, with

d_X = d_Z = 8 witnessed.

Independent 1M-trial re-refutation on two fresh seeds (4271, 9133) the

search agent never used: corroborated on both.

Post-submission re-check (2026-07-27): three more fresh seeds × 8M

gf2_fast trials each, plus one seed × 2M pure-python trials — all four corroborated, weight-8 found every time, nothing lighter.

Filed upper_bound; MILP exact certification not tractable at n = 192.

Dead ends / novelty diligence

Square lattices: strictly dominated at fixed polynomials — the aspect

ratio, not the polynomial search, was the win.

Literature check read the LaTeX sources (not abstracts) of

arXiv:2504.08887, 2504.09171, 2606.19482, 2607.05897; no match. The exhaustive table of 2504.08887 (min n = 264 for k=12, d=8) does not cover this code: its Eq. (9) form needs a single M ∈ GL(2,ℤ) with unit determinant across support differences, and all candidate determinants here are 2,3,5 / 3,4,7 / 5,7,12. Also not a 2BGA (90+90 checks of non-uniform weight, vs 96+96 uniform), so Lin–Pryadko's n ≤ 200 enumeration cannot contain it. novelty: new_parameters remains a submitter claim.

Open leads

The layout is the unoptimized stacked bilayer grid (r = 3.6056). Under g ∝ r⁻⁴ this entry is likely improvable substantially by relayout alone.

Tools

Claude Opus 5 (matches provenance.model), planar-BB campaign of 2026-07-25; boundary_engine.py, gf2_fast for deep RIS, verify/qldpc_verify.py.

Reproduction

research/local2d/boundary_engine.py::build_planar with f = 1 + y + x², g = y + x + x²y on an 8×12 lattice, then reduce_weights; supports and layout in codes/192-12-8.json.

Parity checks

X-checks 90 · Z-checks 90

H_X (90 checks, sparse supports)

[0, 13, 27, 99, 109, 120] [1, 14, 28, 100, 110, 121] [2, 15, 29, 101, 111, 122] [3, 16, 30, 102, 112, 123] [4, 17, 31, 103, 113, 124] [5, 18, 32, 104, 114, 125] [6, 19, 33, 105, 115, 126] [7, 20, 34, 106, 116, 127] [8, 21, 35, 107, 117, 128] [12, 25, 39, 111, 121, 132] [13, 26, 40, 112, 122, 133] [14, 27, 41, 113, 123, 134] [15, 28, 42, 114, 124, 135] [16, 29, 43, 115, 125, 136] [17, 30, 44, 116, 126, 137] [18, 31, 45, 117, 127, 138] [19, 32, 46, 118, 128, 139] [20, 33, 47, 119, 129, 140] [24, 37, 51, 123, 133, 144] [25, 38, 52, 124, 134, 145] [26, 39, 53, 125, 135, 146] [27, 40, 54, 126, 136, 147] [28, 41, 55, 127, 137, 148] [29, 42, 56, 128, 138, 149] [30, 43, 57, 129, 139, 150] [31, 44, 58, 130, 140, 151] [32, 45, 59, 131, 141, 152] [36, 49, 63, 135, 145, 156] [37, 50, 64, 136, 146, 157] [38, 51, 65, 137, 147, 158] [39, 52, 66, 138, 148, 159] [40, 53, 67, 139, 149, 160] [41, 54, 68, 140, 150, 161] [42, 55, 69, 141, 151, 162] [43, 56, 70, 142, 152, 163] [44, 57, 71, 143, 153, 164] [48, 61, 75, 147, 157, 168] [49, 62, 76, 148, 158, 169] [50, 63, 77, 149, 159, 170] [51, 64, 78, 150, 160, 171] [52, 65, 79, 151, 161, 172] [53, 66, 80, 152, 162, 173] [54, 67, 81, 153, 163, 174] [55, 68, 82, 154, 164, 175] [56, 69, 83, 155, 165, 176] [60, 73, 87, 159, 169, 180] [61, 74, 88, 160, 170, 181] [62, 75, 89, 161, 171, 182] [63, 76, 90, 162, 172, 183] [64, 77, 91, 163, 173, 184] [65, 78, 92, 164, 174, 185] [66, 79, 93, 165, 175, 186] [67, 80, 94, 166, 176, 187] [68, 81, 95, 167, 177, 188] [3, 96] [4, 97] [5, 98] [6, 99] [7, 100] [8, 101] [9, 102] [10, 103] [11, 104] [1, 15, 97, 108] [2, 16, 98, 109] [3, 17, 99, 110] [4, 18, 100, 111] [5, 19, 101, 112] [6, 20, 102, 113] [7, 21, 103, 114] [8, 22, 104, 115] [9, 23, 105, 116] [72, 85, 171, 181] [73, 86, 172, 182] [74, 87, 173, 183] [75, 88, 174, 184] [76, 89, 175, 185] [77, 90, 176, 186] [78, 91, 177, 187] [79, 92, 178, 188] [80, 93, 179, 189] [84, 183] [85, 184] [86, 185] [87, 186] [88, 187] [89, 188] [90, 189] [91, 190] [92, 191]

H_Z (90 checks, sparse supports)

[3, 14, 24, 96, 110, 123] [4, 15, 25, 97, 111, 124] [5, 16, 26, 98, 112, 125] [6, 17, 27, 99, 113, 126] [7, 18, 28, 100, 114, 127] [8, 19, 29, 101, 115, 128] [9, 20, 30, 102, 116, 129] [10, 21, 31, 103, 117, 130] [11, 22, 32, 104, 118, 131] [15, 26, 36, 108, 122, 135] [16, 27, 37, 109, 123, 136] [17, 28, 38, 110, 124, 137] [18, 29, 39, 111, 125, 138] [19, 30, 40, 112, 126, 139] [20, 31, 41, 113, 127, 140] [21, 32, 42, 114, 128, 141] [22, 33, 43, 115, 129, 142] [23, 34, 44, 116, 130, 143] [27, 38, 48, 120, 134, 147] [28, 39, 49, 121, 135, 148] [29, 40, 50, 122, 136, 149] [30, 41, 51, 123, 137, 150] [31, 42, 52, 124, 138, 151] [32, 43, 53, 125, 139, 152] [33, 44, 54, 126, 140, 153] [34, 45, 55, 127, 141, 154] [35, 46, 56, 128, 142, 155] [39, 50, 60, 132, 146, 159] [40, 51, 61, 133, 147, 160] [41, 52, 62, 134, 148, 161] [42, 53, 63, 135, 149, 162] [43, 54, 64, 136, 150, 163] [44, 55, 65, 137, 151, 164] [45, 56, 66, 138, 152, 165] [46, 57, 67, 139, 153, 166] [47, 58, 68, 140, 154, 167] [51, 62, 72, 144, 158, 171] [52, 63, 73, 145, 159, 172] [53, 64, 74, 146, 160, 173] [54, 65, 75, 147, 161, 174] [55, 66, 76, 148, 162, 175] [56, 67, 77, 149, 163, 176] [57, 68, 78, 150, 164, 177] [58, 69, 79, 151, 165, 178] [59, 70, 80, 152, 166, 179] [63, 74, 84, 156, 170, 183] [64, 75, 85, 157, 171, 184] [65, 76, 86, 158, 172, 185] [66, 77, 87, 159, 173, 186] [67, 78, 88, 160, 174, 187] [68, 79, 89, 161, 175, 188] [69, 80, 90, 162, 176, 189] [70, 81, 91, 163, 177, 190] [71, 82, 92, 164, 178, 191] [0, 120] [12, 132] [24, 144] [36, 156] [48, 168] [60, 180] [2, 13, 109, 122] [14, 25, 121, 134] [26, 37, 133, 146] [38, 49, 145, 158] [50, 61, 157, 170] [62, 73, 169, 182] [1, 12, 108, 121] [13, 24, 120, 133] [25, 36, 132, 145] [37, 48, 144, 157] [49, 60, 156, 169] [61, 72, 168, 181] [35, 107] [47, 119] [34, 106] [46, 118] [58, 130] [70, 142] [82, 154] [94, 166] [23, 33, 105, 119] [59, 131] [71, 143] [83, 155] [95, 167] [83, 93, 165, 179] [35, 45, 59, 117] [47, 57, 71, 129] [59, 69, 83, 141] [71, 81, 95, 153]

raw submission JSON