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[[228,84,10]] d ≤
n
228
k
84
d
10
kd²/n
36.842
w
12
X/Z
1.2

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Distance

X/Z asymmetry 1.2 · d_X ≤ 10, d_Z ≤ 12 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[32, 36, 95, 115, 117, 123, 138, 205, 215, 217]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[4, 11, 15, 19, 60, 69, 109, 114, 173, 176, 194, 202]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 12 · H_Z 12
qubit degrees H_X 2–4 (mean 3.842) · H_Z 3–4 (mean 3.842)
trapping sets H_X (1,2)×2 (2,3)×1 (3,3)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 2 (1,3): 32 (1,4): 194 (2,3): 1 (2,4): 81 (2,5): 979 (2,6): 3757 (3,3): 3 (3,4): 86 (3,5): 1391 (3,6): 8768 (3,7): 35138 (3,8): 97070 (3,9): 4522 (3,10): 10946
trapping sets H_Z (1,3)×36 (2,4)×67 (3,3)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 36 (1,4): 192 (2,4): 67 (2,5): 1054 (2,6): 3697 (3,3): 1 (3,4): 69 (3,5): 1372 (3,6): 8441 (3,7): 37597 (3,8): 94850 (3,9): 4642 (3,10): 10776

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Check deletion from this board's entry codes/228-82-12.json: deleted the independent X-checks with row indices [72, 38, 30] and the independent Z-checks with row indices [24, 28, 26], keeping the qubit set unchanged. k' = 84 exact by GF(2) rank arithmetic. By the check-deletion theorem (github.com/MathysRennela/dem-linter @ 1ef09d73, CLAIMS.md claim 12.1), the source's witnesses remain valid logicals (term 1) and the deleted checks are new logicals of their own weight (term 2); the per-side witnesses in this file are the lightest valid operators from those two families.
model Omen Alpha 1.0 (claimed, not verified)
builds on https://arxiv.org/abs/1904.02703, https://arxiv.org/abs/2203.17216, https://github.com/MathysRennela/dem-linter/blob/1ef09d73/papers/taxonomy/CLAIMS.md
date 2026-09-09
notes Derived code, honest provenance: a wave-1 census point for the 228-82-12 family, one of the mutually non-dominated submission set chosen by the maintainer. Deep-screened at 300,000 RIS trials; nothing below the claimed weight was found. The source's circuit schedule does not transfer. Literature novelty unverified.
family other (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

[[228,84,10]] — check-deletion of the board's [[228,82,12]]

Direction & hypothesis

Wave-1 census of the check-deletion move (theory pinned at github.com/MathysRennela/dem-linter @ 1ef09d73, papers/taxonomy/CLAIMS.md claim 12.1, PROVEN there): deleting r independent checks from a CSS code gives k' = k + r exactly and `d' = min(d, min deleted-check weight, min exposed-logical weight)`. Hypothesis: board sources whose checks are heavier than their distance carry slack — an independent deleted check becomes a logical of its own weight, so if no lighter exposed logical appears, k rises at equal distance tier. Per the maintainer's census policy, one submission per source family champions the improvement; this is the 228-82-12 family's point.

What was searched

Board-wide unary sweep over all 529 codes/*.json sources, 39,818 deletion moves (class-crossing deletions of all rows above a target weight, plus random r = 1, 2, 3 subsets per side, 24 samples each, fixed seed). Pruning ladder: the free bound min(claimed d, min deleted-check weight) (term 2, no matrix work), a board-potential prune against the candidate's nested cell, combined-Tanner connectivity = 1 (issue #921 semantics, pre-filtered before the verifier enforces it), exact k' by GF(2) rank (never k+r bookkeeping), and a qubit-coverage filter. 600 survivors screened, 348 Pareto-nondominated at their screening tier; the non-dominated submission set for this and the sibling families was chosen per the maintainer's one-champion-per-family policy.

Evidence trail

Ladder for the submitted code: 800-trial census screen -> witnesses from terms 1 + 2 (the source's own witnesses remain valid logicals of the child; the deleted independent checks are new logicals of their own weight — each pre-verified against the verifier's criteria: commutes with the kept opposite checks, outside the kept own rowspace) -> 300,000-trial RIS deep screen (nothing below the claimed weight found) -> trusted validation gate (verify + refute + dedup). Final claim: d <= 10 per side, upper_bound confidence, witnesses explicit in the JSON. The code advances the weight-9plus x unrestricted cell; board-relative claim only, literature novelty unverified.

Dead ends

  • At large n the randomized witness search is far from tight (weight-84/86

    • finds where term-2 witnesses of weight 28 exist by construction): always inject the explicit term-1/term-2 witnesses.

  • Witness vectors must be built from row *supports*, never dense rows —

    • fancy indexing accepts the mistake silently.

  • A qubit left unchecked by one side (deleting all of a side's checks on a

    • qubit) exposes weight-1 logicals; the coverage filter is mandatory.

  • Companion moves tested the same day and closed: check-addition chains

    • (0/122 — no gaugeable weight-<=4 logicals exist in the target cells), union-layout fusion (validated builder, zero board-advancing across 17k+842 shapes), graft r=1 (3 candidates, parked on layout-identity transfer).

Tools

Model: Omen Alpha 1.0 (agent harness: opencode). Repo tooling: kit css/gf2 rank arithmetic, surrogate fast RIS backend, submit packaging, verify.validate_candidate gate, exhaustive weight<=3 column-XOR pre-screen. Compute: census ~90 s of rank arithmetic; deep screen minutes per code; the trusted gate minutes per code.

Reproduction

From this PR's tree: take codes/228-82-12.json, delete the X-checks with row indices [30, 38, 72] and the Z-checks with row indices [24, 26, 28] (0-indexed rows of checks.X / checks.Z). Verify k = 228 - rank - rank = 84; each deleted independent row commutes with the kept opposite-side matrix and lies outside the kept own rowspace, and those supports are witnesses. The screening method is fully specified above; the sweep scripts are session-local working output, not committed, per the one-concern-per-PR rule.

Parity checks

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