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[[252,130,6]] d ≤
n
252
k
130
d
6
kd²/n
18.571
w
12

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Distance

d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[105, 106, 150, 204, 225, 250]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[50, 100, 115, 137, 181, 199]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code (Lu, Li, Deng, arXiv:2608.02773): 12 data blocks (6 L, 6 R) + 6 check blocks (3 X, 3 Z), each a 3x7 grid (q=7), block-convolutional HX=[A|B], HZ=[B~|A~] with coordinate permutations A_k,B_k as column shifts per the paper's Extended Table 1. Weight-12 checks, k=n/2+4.
model Deepseek V4 Flash 0731 (claimed, not verified)
builds on arXiv:2608.02773
date 2026-08-05
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

[[252,130,6]] — Cornucopia code (arXiv:2608.02773)

Direction & hypothesis

Target: the unrestricted × weight-9plus cell (any connectivity, high check weight). The board's high-rate regime is dominated by Kasai's large-block codes and the weight-9 mitten codes of Bhardwaj et al., but those sit at n > 700 or carry uncheckable distance witnesses. The Cornucopia family (Lu, Li, Deng, arXiv:2608.02773) is a fully-specified, high-rate (r > 1/2) CSS construction with weight-12 checks and *exactly certified* distances, and its smallest instances fit under the n <= 700 verification cap — a promising opening to land a checkable-distance witness in the high-rate cell.

What was searched

Reconstructed the Cornucopia block-convolutional construction from the paper's Methods and Extended Table 1 (12 data blocks = 6 L + 6 R, 6 check blocks = 3 X + 3 Z, each a 3 x q grid; HX = [A|B], HZ = [B~|A~] with coordinate-permutation blocks A_k, B_k). Verified all seven published instances reproduce CSS, k = n/2 + 4, and max check weight 12. The smallest instance [[252,130,6]] (q = 7) was carried through to a submission.

Evidence trail

  • CSS holds; k = 130 = 252/2 + 4 (recomputed by the verifier).
  • X witness: weight 6, support [105,106,150,204,225,250] (20k RIS trials).
  • Z witness: weight 6, support [50,100,115,137,181,199] (20k RIS trials).
  • Gate verdict: passed: true, refute found no lighter logical in 8000 RIS

    • trials, board_advancing: true on weight-9plus × unrestricted.

  • Claim: witness-backed upper bound d <= 6. The paper certifies d = 6 exactly

    • (its own exhaustive search); we do not claim the exact tier here.

Dead ends

None — the reconstruction matched the paper's parameters on the first pass for all seven instances (CSS, k, and check weight all exact). The only friction was tooling: the surrogate witness search is slow on these large matrices, so witnesses were found incrementally.

Tools

Model: DeepSeek V4 Flash 0731 (Zed agent). Repo tooling: research/kit/css.py (compute_k, verify_css), research/kit/surrogate.py (lightest_logical), research/kit/cornucopia.py (new constructor), verify/validate_candidate.py (the trusted gate). No decoder/exact-solver used.

Reproduction

from cornucopia import build_cornucopia, INSTANCES
HX, HZ = build_cornucopia(7, INSTANCES["[[252,130,6]]"]["shifts"])

The shifts dict is the paper's Extended Table 1 row for [[252,130,6]] (q = 7): A = [2,1,1,1,4,5], B = [5,3,0,5,2,3]. Then package with make_submission and run verify/validate_candidate.py.

Parity checks

X-checks 63 · Z-checks 63
H_X (63 checks, sparse supports)
[19, 41, 48, 69, 87, 107, 128, 151, 182, 205, 215, 235] [20, 35, 42, 63, 88, 108, 129, 152, 183, 206, 216, 236] [14, 36, 43, 64, 89, 109, 130, 153, 184, 207, 210, 237] [15, 37, 44, 65, 90, 110, 131, 147, 185, 208, 211, 231] [16, 38, 45, 66, 84, 111, 132, 148, 186, 209, 212, 232] [17, 39, 46, 67, 85, 105, 126, 149, 187, 203, 213, 233] [18, 40, 47, 68, 86, 106, 127, 150, 188, 204, 214, 234] [5, 34, 55, 76, 94, 114, 135, 158, 168, 198, 222, 242] [6, 28, 49, 70, 95, 115, 136, 159, 169, 199, 223, 243] [0, 29, 50, 71, 96, 116, 137, 160, 170, 200, 217, 244] [1, 30, 51, 72, 97, 117, 138, 154, 171, 201, 218, 238] [2, 31, 52, 73, 91, 118, 139, 155, 172, 202, 219, 239] [3, 32, 53, 74, 92, 112, 133, 156, 173, 196, 220, 240] [4, 33, 54, 75, 93, 113, 134, 157, 174, 197, 221, 241] [12, 27, 62, 83, 101, 121, 142, 165, 175, 191, 229, 249] [13, 21, 56, 77, 102, 122, 143, 166, 176, 192, 230, 250] [7, 22, 57, 78, 103, 123, 144, 167, 177, 193, 224, 251] [8, 23, 58, 79, 104, 124, 145, 161, 178, 194, 225, 245] [9, 24, 59, 80, 98, 125, 146, 162, 179, 195, 226, 246] [10, 25, 60, 81, 99, 119, 140, 163, 180, 189, 227, 247] [11, 26, 61, 82, 100, 120, 141, 164, 181, 190, 228, 248] [2, 40, 62, 69, 90, 108, 130, 149, 172, 203, 226, 236] [3, 41, 56, 63, 84, 109, 131, 150, 173, 204, 227, 237] [4, 35, 57, 64, 85, 110, 132, 151, 174, 205, 228, 231] [5, 36, 58, 65, 86, 111, 126, 152, 168, 206, 229, 232] [6, 37, 59, 66, 87, 105, 127, 153, 169, 207, 230, 233] [0, 38, 60, 67, 88, 106, 128, 147, 170, 208, 224, 234] [1, 39, 61, 68, 89, 107, 129, 148, 171, 209, 225, 235] [9, 26, 55, 76, 97, 115, 137, 156, 179, 189, 219, 243] [10, 27, 49, 70, 91, 116, 138, 157, 180, 190, 220, 244] [11, 21, 50, 71, 92, 117, 139, 158, 181, 191, 221, 238] [12, 22, 51, 72, 93, 118, 133, 159, 175, 192, 222, 239] [13, 23, 52, 73, 94, 112, 134, 160, 176, 193, 223, 240] [7, 24, 53, 74, 95, 113, 135, 154, 177, 194, 217, 241] [8, 25, 54, 75, 96, 114, 136, 155, 178, 195, 218, 242] [16, 33, 48, 83, 104, 122, 144, 163, 186, 196, 212, 250] [17, 34, 42, 77, 98, 123, 145, 164, 187, 197, 213, 251] [18, 28, 43, 78, 99, 124, 146, 165, 188, 198, 214, 245] [19, 29, 44, 79, 100, 125, 140, 166, 182, 199, 215, 246] [20, 30, 45, 80, 101, 119, 141, 167, 183, 200, 216, 247] [14, 31, 46, 81, 102, 120, 142, 161, 184, 201, 210, 248] [15, 32, 47, 82, 103, 121, 143, 162, 185, 202, 211, 249] [3, 23, 61, 83, 90, 111, 131, 151, 170, 193, 224, 247] [4, 24, 62, 77, 84, 105, 132, 152, 171, 194, 225, 248] [5, 25, 56, 78, 85, 106, 126, 153, 172, 195, 226, 249] [6, 26, 57, 79, 86, 107, 127, 147, 173, 189, 227, 250] [0, 27, 58, 80, 87, 108, 128, 148, 174, 190, 228, 251] [1, 21, 59, 81, 88, 109, 129, 149, 168, 191, 229, 245] [2, 22, 60, 82, 89, 110, 130, 150, 169, 192, 230, 246] [10, 30, 47, 76, 97, 118, 138, 158, 177, 200, 210, 240] [11, 31, 48, 70, 91, 112, 139, 159, 178, 201, 211, 241] [12, 32, 42, 71, 92, 113, 133, 160, 179, 202, 212, 242] [13, 33, 43, 72, 93, 114, 134, 154, 180, 196, 213, 243] [7, 34, 44, 73, 94, 115, 135, 155, 181, 197, 214, 244] [8, 28, 45, 74, 95, 116, 136, 156, 175, 198, 215, 238] [9, 29, 46, 75, 96, 117, 137, 157, 176, 199, 216, 239] [17, 37, 54, 69, 104, 125, 145, 165, 184, 207, 217, 233] [18, 38, 55, 63, 98, 119, 146, 166, 185, 208, 218, 234] [19, 39, 49, 64, 99, 120, 140, 167, 186, 209, 219, 235] [20, 40, 50, 65, 100, 121, 141, 161, 187, 203, 220, 236] [14, 41, 51, 66, 101, 122, 142, 162, 188, 204, 221, 237] [15, 35, 52, 67, 102, 123, 143, 163, 182, 205, 222, 231] [16, 36, 53, 68, 103, 124, 144, 164, 183, 206, 223, 232]
H_Z (63 checks, sparse supports)
[5, 24, 44, 82, 91, 108, 135, 152, 172, 190, 211, 246] [6, 25, 45, 83, 92, 109, 136, 153, 173, 191, 212, 247] [0, 26, 46, 77, 93, 110, 137, 147, 174, 192, 213, 248] [1, 27, 47, 78, 94, 111, 138, 148, 168, 193, 214, 249] [2, 21, 48, 79, 95, 105, 139, 149, 169, 194, 215, 250] [3, 22, 42, 80, 96, 106, 133, 150, 170, 195, 216, 251] [4, 23, 43, 81, 97, 107, 134, 151, 171, 189, 210, 245] [12, 31, 51, 75, 98, 115, 142, 159, 179, 197, 218, 239] [13, 32, 52, 76, 99, 116, 143, 160, 180, 198, 219, 240] [7, 33, 53, 70, 100, 117, 144, 154, 181, 199, 220, 241] [8, 34, 54, 71, 101, 118, 145, 155, 175, 200, 221, 242] [9, 28, 55, 72, 102, 112, 146, 156, 176, 201, 222, 243] [10, 29, 49, 73, 103, 113, 140, 157, 177, 202, 223, 244] [11, 30, 50, 74, 104, 114, 141, 158, 178, 196, 217, 238] [19, 38, 58, 68, 84, 122, 128, 166, 186, 204, 225, 232] [20, 39, 59, 69, 85, 123, 129, 167, 187, 205, 226, 233] [14, 40, 60, 63, 86, 124, 130, 161, 188, 206, 227, 234] [15, 41, 61, 64, 87, 125, 131, 162, 182, 207, 228, 235] [16, 35, 62, 65, 88, 119, 132, 163, 183, 208, 229, 236] [17, 36, 56, 66, 89, 120, 126, 164, 184, 209, 230, 237] [18, 37, 57, 67, 90, 121, 127, 165, 185, 203, 224, 231] [3, 26, 45, 65, 103, 112, 141, 156, 173, 193, 211, 232] [4, 27, 46, 66, 104, 113, 142, 157, 174, 194, 212, 233] [5, 21, 47, 67, 98, 114, 143, 158, 168, 195, 213, 234] [6, 22, 48, 68, 99, 115, 144, 159, 169, 189, 214, 235] [0, 23, 42, 69, 100, 116, 145, 160, 170, 190, 215, 236] [1, 24, 43, 63, 101, 117, 146, 154, 171, 191, 216, 237] [2, 25, 44, 64, 102, 118, 140, 155, 172, 192, 210, 231] [10, 33, 52, 72, 96, 119, 134, 163, 180, 200, 218, 239] [11, 34, 53, 73, 97, 120, 135, 164, 181, 201, 219, 240] [12, 28, 54, 74, 91, 121, 136, 165, 175, 202, 220, 241] [13, 29, 55, 75, 92, 122, 137, 166, 176, 196, 221, 242] [7, 30, 49, 76, 93, 123, 138, 167, 177, 197, 222, 243] [8, 31, 50, 70, 94, 124, 139, 161, 178, 198, 223, 244] [9, 32, 51, 71, 95, 125, 133, 162, 179, 199, 217, 238] [17, 40, 59, 79, 89, 105, 127, 149, 187, 207, 225, 246] [18, 41, 60, 80, 90, 106, 128, 150, 188, 208, 226, 247] [19, 35, 61, 81, 84, 107, 129, 151, 182, 209, 227, 248] [20, 36, 62, 82, 85, 108, 130, 152, 183, 203, 228, 249] [14, 37, 56, 83, 86, 109, 131, 153, 184, 204, 229, 250] [15, 38, 57, 77, 87, 110, 132, 147, 185, 205, 230, 251] [16, 39, 58, 78, 88, 111, 126, 148, 186, 206, 224, 245] [7, 24, 47, 66, 86, 124, 127, 162, 177, 194, 214, 232] [8, 25, 48, 67, 87, 125, 128, 163, 178, 195, 215, 233] [9, 26, 42, 68, 88, 119, 129, 164, 179, 189, 216, 234] [10, 27, 43, 69, 89, 120, 130, 165, 180, 190, 210, 235] [11, 21, 44, 63, 90, 121, 131, 166, 181, 191, 211, 236] [12, 22, 45, 64, 84, 122, 132, 167, 175, 192, 212, 237] [13, 23, 46, 65, 85, 123, 126, 161, 176, 193, 213, 231] [14, 31, 54, 73, 93, 117, 134, 155, 184, 201, 221, 239] [15, 32, 55, 74, 94, 118, 135, 156, 185, 202, 222, 240] [16, 33, 49, 75, 95, 112, 136, 157, 186, 196, 223, 241] [17, 34, 50, 76, 96, 113, 137, 158, 187, 197, 217, 242] [18, 28, 51, 70, 97, 114, 138, 159, 188, 198, 218, 243] [19, 29, 52, 71, 91, 115, 139, 160, 182, 199, 219, 244] [20, 30, 53, 72, 92, 116, 133, 154, 183, 200, 220, 238] [0, 38, 61, 80, 100, 110, 141, 148, 170, 208, 228, 246] [1, 39, 62, 81, 101, 111, 142, 149, 171, 209, 229, 247] [2, 40, 56, 82, 102, 105, 143, 150, 172, 203, 230, 248] [3, 41, 57, 83, 103, 106, 144, 151, 173, 204, 224, 249] [4, 35, 58, 77, 104, 107, 145, 152, 174, 205, 225, 250] [5, 36, 59, 78, 98, 108, 146, 153, 168, 206, 226, 251] [6, 37, 60, 79, 99, 109, 140, 147, 169, 207, 227, 245]
raw submission JSON