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[[242,12,12]] d ≤
n
242
k
12
d
12
kd²/n
7.14
w
8
g
0.0035
r
6.7082
layers
2

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Distance

d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[33, 49, 70, 127, 143, 162, 180, 198, 215, 218, 220, 237]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[23, 28, 33, 40, 50, 57, 59, 69, 72, 113, 184, 186, 200]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 6.708
X checkZ checkqubit site (150)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 @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction open-boundary planar BB (weight-8 tile family); f=[(0,0),(1,0),(2,3),(2,-2)], g=[(0,0),(0,1),(-1,-1),(-1,3)] on 8x19 lattice; boundary_engine.build_planar
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-08-07
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (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

[[242,12,12]] — open-boundary planar bivariate-bicycle (weight-8 tile family), aspect-ratio sweep

Direction & hypothesis

Target: the weight-8 × local-2d-bilayer cell, one of the sparsest on the board. Hypothesis: for open-boundary planar BB codes the distance is set by the shorter lattice axis (per-axis transfer-graph slopes), so at fixed polynomials a *balanced rectangle* spends qubits more efficiently than a square — the same mechanism that produced [[192,12,8]] in the weight-6 cell.

What was searched

A lattice-geometry sweep at fixed weight-8 tile-family supports f = [(0,0),(1,0),(2,3),(2,-2)], g = [(0,0),(0,1),(-1,-1),(-1,3)] (the family of the board's [[294,12,14]] flagship). Enumerated Lx ∈ [6,8], Ly ∈ [10,23] within the engine's validity domain (L ≥ extent + 3) and n ≤ 700. Built via research/local2d/boundary_engine.py::build_planar + reduce_weights; screened with distance_rand (300 trials) for d ≥ 3, deduped against the board by fingerprint. k = 12 is stable across all lattice sizes (equals the mixed volume of the support pair).

Best board-advancing hit: Lx=8, Ly=19 → [[242,12,12]], kd²/n = 7.14. Larger lattices reach higher d (e.g. [[268,12,13]], [[294,12,13]]) but are dominated by existing board entries ([[263,16,12]], [[294,12,14]]); this one is not.

Evidence trail

  • Screening: d ≤ 12 at 300 RIS trials (upper bound).
  • Confirmation ladder: dX ≤ 12, dZ ≤ 13 (min ≤ 12) flat across 400 → 1k → 2k →

    • 10k RIS trials/side — the value does not drop as trials increase, so the d = 12 claim is not inflated.

  • Trusted gate (verify/validate_candidate.py): passed: true,

    • refuted: false (no lighter logical in the fresh-seed RIS refutation), dedup: none (no exact or WL-equivalent board duplicate).

  • board_advancing: true — no dominator in the weight-8 × local-2d-bilayer

    • cell.

  • Filed upper_bound (d_X = 12, d_Z = 13 witnesses); MILP exact

    • certification not attempted at n = 242, k = 12.

Dead ends

  • Square-ish lattices (Lx ≈ Ly) at this support pair give lower efficiency:

    • the aspect ratio, not the polynomial search, was the win.

  • The same family at Lx=6 and Lx=7 produces dominated codes (d ≤ 10 at

    • comparable n); Lx=8 with Ly ≥ 19 is where d reaches 12 without being dominated.

  • Higher-d hits ([[268,12,13]], [[294,12,13]]) are dominated by existing board

    • entries; only this one advances the cell.

Tools

Autoresearch agent (DeepSeek V4 Flash 0731, matches provenance.model; author @MathysRennela), planar-BB population sweep of 2026-08-07; boundary_engine.py, planar.grid_coordinates, research/kit/submit.py, verify/validate_candidate.py.

Reproduction

research/local2d/boundary_engine.py::build_planar(8, 19, S_f, S_g) with S_f = [(0,0),(1,0),(2,3),(2,-2)], S_g = [(0,0),(0,1),(-1,-1),(-1,3)], then reduce_weights; layout via planar.grid_coordinates(8, 19, kept=...), layers=2. Supports and layout in codes/242-12-12.json.

Parity checks

X-checks 157 · Z-checks 138
H_X (157 checks, sparse supports)
[2, 19, 24, 114, 118, 133, 134] [3, 20, 25, 115, 119, 134, 135] [4, 21, 26, 116, 120, 135, 136] [5, 22, 27, 117, 121, 136, 137] [6, 23, 28, 118, 122, 137, 138] [7, 24, 29, 119, 123, 138, 139] [8, 25, 30, 120, 124, 139, 140] [9, 26, 31, 121, 125, 140, 141] [10, 27, 32, 122, 126, 141, 142] [11, 28, 33, 123, 127, 142, 143] [12, 29, 34, 124, 128, 143, 144] [13, 30, 35, 125, 129, 144, 145] [14, 31, 36, 126, 130, 145, 146] [15, 32, 37, 127, 131, 146, 147] [2, 21, 38, 43, 132, 136, 151, 152] [3, 22, 39, 44, 133, 137, 152, 153] [4, 23, 40, 45, 134, 138, 153, 154] [5, 24, 41, 46, 135, 139, 154, 155] [6, 25, 42, 47, 136, 140, 155, 156] [7, 26, 43, 48, 137, 141, 156, 157] [8, 27, 44, 49, 138, 142, 157, 158] [9, 28, 45, 50, 139, 143, 158, 159] [10, 29, 46, 51, 140, 144, 159, 160] [11, 30, 47, 52, 141, 145, 160, 161] [12, 31, 48, 53, 142, 146, 161, 162] [13, 32, 49, 54, 143, 147, 162, 163] [14, 33, 50, 55, 144, 148, 163, 164] [15, 34, 51, 56, 145, 149, 164, 165] [21, 40, 57, 62, 150, 154, 169, 170] [22, 41, 58, 63, 151, 155, 170, 171] [23, 42, 59, 64, 152, 156, 171, 172] [24, 43, 60, 65, 153, 157, 172, 173] [25, 44, 61, 66, 154, 158, 173, 174] [26, 45, 62, 67, 155, 159, 174, 175] [27, 46, 63, 68, 156, 160, 175, 176] [28, 47, 64, 69, 157, 161, 176, 177] [29, 48, 65, 70, 158, 162, 177, 178] [30, 49, 66, 71, 159, 163, 178, 179] [31, 50, 67, 72, 160, 164, 179, 180] [32, 51, 68, 73, 161, 165, 180, 181] [33, 52, 69, 74, 162, 166, 181, 182] [34, 53, 70, 75, 163, 167, 182, 183] [40, 59, 76, 81, 168, 172, 187, 188] [41, 60, 77, 82, 169, 173, 188, 189] [42, 61, 78, 83, 170, 174, 189, 190] [43, 62, 79, 84, 171, 175, 190, 191] [44, 63, 80, 85, 172, 176, 191, 192] [45, 64, 81, 86, 173, 177, 192, 193] [46, 65, 82, 87, 174, 178, 193, 194] [47, 66, 83, 88, 175, 179, 194, 195] [48, 67, 84, 89, 176, 180, 195, 196] [49, 68, 85, 90, 177, 181, 196, 197] [50, 69, 86, 91, 178, 182, 197, 198] [51, 70, 87, 92, 179, 183, 198, 199] [52, 71, 88, 93, 180, 184, 199, 200] [53, 72, 89, 94, 181, 185, 200, 201] [59, 78, 95, 100, 186, 190, 206, 207] [60, 79, 96, 101, 187, 191, 207, 208] [61, 80, 97, 102, 188, 192, 208, 209] [62, 81, 98, 103, 189, 193, 209, 210] [63, 82, 99, 104, 190, 194, 210, 211] [64, 83, 100, 105, 191, 195, 211, 212] [65, 84, 101, 106, 192, 196, 212, 213] [66, 85, 102, 107, 193, 197, 213, 214] [67, 86, 103, 108, 194, 198, 214, 215] [68, 87, 104, 109, 195, 199, 215, 216] [69, 88, 105, 110, 196, 200, 216, 217] [70, 89, 106, 111, 197, 201, 217, 218] [71, 90, 107, 112, 198, 202, 218, 219] [72, 91, 108, 113, 199, 203, 219, 220] [0, 1, 2, 3, 4, 115] [1, 2, 3, 4, 5, 116] [2, 3, 4, 5, 6, 117] [3, 4, 5, 6, 7, 118] [4, 5, 6, 7, 8, 119] [5, 6, 7, 8, 9, 120] [6, 7, 8, 9, 10, 121] [7, 8, 9, 10, 11, 122] [8, 9, 10, 11, 12, 123] [9, 10, 11, 12, 13, 124] [10, 11, 12, 13, 14, 125] [11, 12, 13, 14, 15, 126] [12, 13, 14, 15, 16, 127] [13, 14, 15, 16, 17, 128] [14, 15, 16, 17, 18, 129] [2, 19, 24, 114, 118, 133, 134] [3, 20, 25, 115, 119, 134, 135] [4, 21, 26, 116, 120, 135, 136] [5, 22, 27, 117, 121, 136, 137] [6, 23, 28, 118, 122, 137, 138] [7, 24, 29, 119, 123, 138, 139] [8, 25, 30, 120, 124, 139, 140] [9, 26, 31, 121, 125, 140, 141] [10, 27, 32, 122, 126, 141, 142] [11, 28, 33, 123, 127, 142, 143] [12, 29, 34, 124, 128, 143, 144] [13, 30, 35, 125, 129, 144, 145] [14, 31, 36, 126, 130, 145, 146] [15, 32, 37, 127, 131, 146, 147] [0, 5, 115, 116] [1, 6, 116, 117] [2, 7, 117, 118] [3, 8, 118, 119] [4, 9, 119, 120] [5, 10, 120, 121] [6, 11, 121, 122] [7, 12, 122, 123] [8, 13, 123, 124] [9, 14, 124, 125] [10, 15, 125, 126] [11, 16, 126, 127] [12, 17, 127, 128] [13, 18, 128, 129] [97, 224, 228] [98, 225, 229] [99, 226, 230] [100, 227, 231] [101, 228, 232] [102, 229, 233] [103, 230, 234] [104, 231, 235] [105, 232, 236] [106, 233, 237] [107, 234, 238] [108, 235, 239] [109, 236, 240] [110, 237, 241] [78, 97, 205, 209, 225, 226] [79, 98, 206, 210, 226, 227] [80, 99, 207, 211, 227, 228] [81, 100, 208, 212, 228, 229] [82, 101, 209, 213, 229, 230] [83, 102, 210, 214, 230, 231] [84, 103, 211, 215, 231, 232] [85, 104, 212, 216, 232, 233] [86, 105, 213, 217, 233, 234] [87, 106, 214, 218, 234, 235] [88, 107, 215, 219, 235, 236] [89, 108, 216, 220, 236, 237] [90, 109, 217, 221, 237, 238] [91, 110, 218, 222, 238, 239] [97, 224, 228] [98, 225, 229] [99, 226, 230] [100, 227, 231] [101, 228, 232] [102, 229, 233] [103, 230, 234] [104, 231, 235] [105, 232, 236] [106, 233, 237] [107, 234, 238] [108, 235, 239] [109, 236, 240] [110, 237, 241] [77, 96, 204, 208, 224, 225] [96, 223, 227]
H_Z (138 checks, sparse supports)
[2, 3, 19, 23, 118, 134, 152] [3, 4, 20, 24, 114, 119, 135, 153] [4, 5, 21, 25, 115, 120, 136, 154] [5, 6, 22, 26, 116, 121, 137, 155] [6, 7, 23, 27, 117, 122, 138, 156] [7, 8, 24, 28, 118, 123, 139, 157] [8, 9, 25, 29, 119, 124, 140, 158] [9, 10, 26, 30, 120, 125, 141, 159] [10, 11, 27, 31, 121, 126, 142, 160] [11, 12, 28, 32, 122, 127, 143, 161] [12, 13, 29, 33, 123, 128, 144, 162] [13, 14, 30, 34, 124, 129, 145, 163] [14, 15, 31, 35, 125, 130, 146, 164] [15, 16, 32, 36, 126, 131, 147, 165] [21, 22, 38, 42, 136, 152, 170] [22, 23, 39, 43, 132, 137, 153, 171] [23, 24, 40, 44, 133, 138, 154, 172] [24, 25, 41, 45, 134, 139, 155, 173] [25, 26, 42, 46, 135, 140, 156, 174] [26, 27, 43, 47, 136, 141, 157, 175] [27, 28, 44, 48, 137, 142, 158, 176] [28, 29, 45, 49, 138, 143, 159, 177] [29, 30, 46, 50, 139, 144, 160, 178] [30, 31, 47, 51, 140, 145, 161, 179] [31, 32, 48, 52, 141, 146, 162, 180] [32, 33, 49, 53, 142, 147, 163, 181] [33, 34, 50, 54, 143, 148, 164, 182] [34, 35, 51, 55, 144, 149, 165, 183] [40, 41, 57, 61, 154, 170, 188] [41, 42, 58, 62, 150, 155, 171, 189] [42, 43, 59, 63, 151, 156, 172, 190] [43, 44, 60, 64, 152, 157, 173, 191] [44, 45, 61, 65, 153, 158, 174, 192] [45, 46, 62, 66, 154, 159, 175, 193] [46, 47, 63, 67, 155, 160, 176, 194] [47, 48, 64, 68, 156, 161, 177, 195] [48, 49, 65, 69, 157, 162, 178, 196] [49, 50, 66, 70, 158, 163, 179, 197] [50, 51, 67, 71, 159, 164, 180, 198] [51, 52, 68, 72, 160, 165, 181, 199] [52, 53, 69, 73, 161, 166, 182, 200] [53, 54, 70, 74, 162, 167, 183, 201] [59, 60, 76, 80, 172, 188, 207] [60, 61, 77, 81, 168, 173, 189, 208] [61, 62, 78, 82, 169, 174, 190, 209] [62, 63, 79, 83, 170, 175, 191, 210] [63, 64, 80, 84, 171, 176, 192, 211] [64, 65, 81, 85, 172, 177, 193, 212] [65, 66, 82, 86, 173, 178, 194, 213] [66, 67, 83, 87, 174, 179, 195, 214] [67, 68, 84, 88, 175, 180, 196, 215] [68, 69, 85, 89, 176, 181, 197, 216] [69, 70, 86, 90, 177, 182, 198, 217] [70, 71, 87, 91, 178, 183, 199, 218] [71, 72, 88, 92, 179, 184, 200, 219] [72, 73, 89, 93, 180, 185, 201, 220] [78, 79, 95, 99, 190, 207, 226] [79, 80, 96, 100, 186, 191, 208, 227] [80, 81, 97, 101, 187, 192, 209, 228] [81, 82, 98, 102, 188, 193, 210, 229] [82, 83, 99, 103, 189, 194, 211, 230] [83, 84, 100, 104, 190, 195, 212, 231] [84, 85, 101, 105, 191, 196, 213, 232] [85, 86, 102, 106, 192, 197, 214, 233] [86, 87, 103, 107, 193, 198, 215, 234] [87, 88, 104, 108, 194, 199, 216, 235] [88, 89, 105, 109, 195, 200, 217, 236] [89, 90, 106, 110, 196, 201, 218, 237] [90, 91, 107, 111, 197, 202, 219, 238] [91, 92, 108, 112, 198, 203, 220, 239] [19, 114] [38, 132] [57, 150] [76, 168] [95, 186] [0, 20, 115] [19, 39, 133] [38, 58, 151] [0, 1, 21, 116, 132, 150] [19, 20, 40, 134, 150, 168] [38, 39, 59, 152, 168, 186] [57, 58, 78, 170, 186, 205] [76, 77, 97, 188, 205, 224] [1, 2, 22, 117, 133, 151] [20, 21, 41, 135, 151, 169] [39, 40, 60, 153, 169, 187] [58, 59, 79, 171, 187, 206] [77, 78, 98, 189, 206, 225] [2, 3, 19, 23, 118, 134, 152] [21, 22, 38, 42, 136, 152, 170] [40, 41, 57, 61, 154, 170, 188] [59, 60, 76, 80, 172, 188, 207] [78, 79, 95, 99, 190, 207, 226] [3, 4, 19, 20, 24, 119, 135, 153] [22, 23, 38, 39, 43, 137, 153, 171] [41, 42, 57, 58, 62, 155, 171, 189] [60, 61, 76, 77, 81, 173, 189, 208] [79, 80, 95, 96, 100, 191, 208, 227] [0, 20, 115] [19, 39, 133] [38, 58, 151] [57, 77, 169, 204] [76, 96, 187, 204, 223] [37, 131] [56, 149] [75, 167] [94, 185] [113, 203] [36, 130] [55, 148] [74, 166] [93, 184] [112, 202] [18, 35, 129] [37, 54, 147] [56, 73, 165] [75, 92, 183] [94, 111, 201] [17, 18, 34, 128, 149, 167] [36, 37, 53, 146, 167, 185] [55, 56, 72, 164, 185, 203] [74, 75, 91, 182, 203, 222] [93, 94, 110, 200, 222, 241] [16, 17, 33, 37, 127, 148, 166] [35, 36, 52, 56, 145, 166, 184] [54, 55, 71, 75, 163, 184, 202] [73, 74, 90, 94, 181, 202, 221] [92, 93, 109, 113, 199, 221, 240] [15, 16, 32, 36, 37, 126, 147, 165] [34, 35, 51, 55, 56, 144, 165, 183] [53, 54, 70, 74, 75, 162, 183, 201] [72, 73, 89, 93, 94, 180, 201, 220] [91, 92, 108, 112, 113, 198, 220, 239] [14, 15, 31, 35, 36, 125, 146, 164] [33, 34, 50, 54, 55, 143, 164, 182] [52, 53, 69, 73, 74, 161, 182, 200] [71, 72, 88, 92, 93, 179, 200, 219] [90, 91, 107, 111, 112, 197, 219, 238]
raw submission JSON