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[[324,166,6]] d ≤
n
324
k
166
d
6
kd²/n
18.444
w
12
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[30, 79, 87, 132, 158, 236]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[34, 81, 170, 183, 217, 223]
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 4 · H_Z 4 (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 3 · H_Z 3
trapping sets H_X (1,3)×324 (2,2)×81 (3,3)×3249 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 324 (2,2): 81 (2,4): 5184 (3,3): 3249 (3,5): 107775 (3,7): 16245
trapping sets H_Z (1,3)×324 (2,2)×81 (3,3)×3249 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 324 (2,2): 81 (2,4): 5184 (3,3): 3249 (3,5): 107775 (3,7): 16245

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code (Lu, Li & Deng, arXiv:2608.02773; block-convolutional CSS construction with the check matrices given in the Methods of arXiv:2610.19073): q=9, n=36q=324 data qubits in twelve 3x9 tiles L0..L5,R0..R5 plus nine 3x9 X-check and nine 3x9 Z-check tiles; HX=[A_(j-i)|B_(j-i)], HZ=[B-1_(i-j)|A-1_(i-j)] with the coordinate permutations A_p,B_p:(r,c)|->(rho(p,r),c+s[p]) on Z_3 x Z_9, row maps rho_A(0)=r+1, rho_A(1)=2r+2, rho_B(2)=r+1, rho_B(3)=2r+2 (identity otherwise), and column shifts sA=[5, 7, 5, 1, 5, 6], sB=[0, 3, 7, 1, 1, 1]. These shifts were found by a random shift search; the paper publishes instances only at q=7,16,25,29,49,64,79. Weight-12 checks, k=18q+4=166.
model MiMo-V2.6-Flash (claimed, not verified)
builds on arXiv:2608.02773, arXiv:2610.19073
date 2026-10-15
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

[[324,166,6]] - Cornucopia code at q=9, a new member of the published family

Direction & hypothesis

Target: the weight-9plus x unrestricted cell on the CSS board (checks of weight 12, no 2D-local layout). The Cornucopia architecture paper (arXiv:2610.19073) closes by saying its construction "carries over to other codes of the family", and the family (Lu, Li & Deng, arXiv:2608.02773) is n = 36q, k = 18q + 4 with weight-12 checks - but the paper publishes instances only at q = 7, 16, 25, 29, 49, 64, 79, of which only q=7 and q=16 fit this board's n <= 700 cap and both are already board entries. So the unpublished block sizes under the cap, q = 4..19, were the opening: every one of them is a rate-1/2 point no CSS board entry can dominate (an entry dominating [[324,166,d]] would need n <= 324 and k >= 166, i.e. rate at least 0.512 at that size, and the board has none).

What was searched

The family's only free parameters are its 12 column shifts: check matrices HX = [A_(j-i) | B_(j-i)], HZ = [B^-1_(i-j) | A^-1_(i-j)] over Z_3 x Z_q coordinate permutations (row maps r+1 for A0 and B2, 2r+2 for A1 and B3, identity otherwise). CSS commutation holds for *every* shift choice: the only noncommuting row-map pairs, (A0,B3) and (A1,B2), enter block-row differences i'-i = 3 (mod 6), which never occur for i,i' in {0,1,2}; so the search is over shifts alone, with k recomputed exactly on every draw.

At q=9: 1200 uniformly random shift pairs, screened at 100 RIS trials each (accelerated backend); the top 24 re-measured at 4000; the top 4 re-measured at 100000; then the packaged candidate re-verified by submit.make_submission (5000 numpy trials/side) and by the submission CLI (800 python RIS + 300000 accelerated trials/side). Stage-1 reads at q=9: 2 -> 262, 4 -> 788, 6 -> 148. Seed 17 throughout the sweep, 0 for the CLI witness search. One design fact from the sweep: a read is the lightest logical *found*, so it is always >= the true d - stage-1 ranking cannot discard a better code, only fail to see how good it is.

Evidence trail

Ladder for this candidate (fast-RIS read -> numpy witness -> CLI):

  • stage reads at 100, 4000, 100000 fast-RIS trials: 6, 6, 6; packaging numpy witness search: 5000 trials/side -> d <= 6.
  • final claim: witness-backed upper bound d <= 6, not exact (the

    • exact tier needs the server MILP; k = 166 puts this beyond its envelope).

  • k = 166 = 18*9+4 recomputed by the verifier; CSS commutation checked;

    • max check weight 12.

  • Gate (verify/validate_candidate.py): passed: true, refuted: false

    • (its own refutation found no lighter logical), board_advancing: true, d_only_gain: false, dominated_by: [].

  • The 1200 draws at this q never read above 6; deeper stages confirmed

    • 6. kd^2/n = 18.44.

Dead ends

  • ~0.2% of random shift sets drop rank (k < 18q+4): the two

    • redundancies per check type are not always the only ones. Rejected on an exact GF(2) rank, not a heuristic. At q=9: 2 of 1200 draws.

  • At q=7 and q=16 the best random-shift reads (6 and 8) only *equal* the

    • published-instance board entries [[252,130,6]] and [[576,292,8]], so nothing was staged there: a tie is not a submission.

  • Across the whole campaign (q=4..19, 19,200 draws) no candidate anywhere read

    • above 8 at any depth: uniform random shifts do not reach d=9/10 under n <= 700; the published family first reaches d=10 at q=25 (n=900, over cap). A designed search (hill-climbing on the plateau of 8-reads) is the untried rung, a wider uniform sweep is not.

Tools

Model: MiMo-V2.6-Flash (matches provenance.model), driving an OpenCode agent harness. Repo tooling: research/kit/css.py (exact k, CSS check), research/kit/surrogate.py (RIS surrogate, fast backend verify/gf2_fast), research/kit/submit.py packaging, verify/validate_candidate.py (the trusted gate), ./qldpc submit (witness search, verifier, circuit tier). Compute: ~9 CPU-hours for the whole 16-q campaign; this q's share is the 1200+24*4000+4*100000 screened trials above.

Reproduction

Rebuild (H_X, H_Z) exactly: q = 9, n = 36q, data qubit (b,r,c) (index (3b+r)q+c, b = 0..5 tiles L, b = 6..11 tiles R, r in Z_3, c in Z_q); HX[(i,r,c), (j, A_(j-i)(r,c))] = 1 and on the R tile with B; HZ[(i,r,c), (j, B^-1_(i-j)(r,c))] = 1 and on the R tile with A; each A_p, B_p : (r,c) |-> (rho(p,r), c + s[p]) mod q with rho_A(0)=r+1, rho_A(1)=2r+2, rho_B(2)=r+1, rho_B(3)=2r+2, identity otherwise, and shifts sA = [5, 7, 5, 1, 5, 6], sB = [0, 3, 7, 1, 1, 1]. Weight-12 checks, k = 18q + 4. Verify: uv run python verify/qldpc_verify.py codes/324-166-6.json.

Parity checks

X-checks 81 (max weight 12) · Z-checks 81 (max weight 12)
H_X (81 checks, sparse supports)
[14, 52, 59, 82, 113, 141, 162, 192, 232, 262, 271, 298] [15, 53, 60, 83, 114, 142, 163, 193, 233, 263, 272, 299] [16, 45, 61, 84, 115, 143, 164, 194, 225, 264, 273, 300] [17, 46, 62, 85, 116, 135, 165, 195, 226, 265, 274, 301] [9, 47, 54, 86, 108, 136, 166, 196, 227, 266, 275, 302] [10, 48, 55, 87, 109, 137, 167, 197, 228, 267, 276, 303] [11, 49, 56, 88, 110, 138, 168, 189, 229, 268, 277, 304] [12, 50, 57, 89, 111, 139, 169, 190, 230, 269, 278, 305] [13, 51, 58, 81, 112, 140, 170, 191, 231, 261, 270, 297] [23, 43, 68, 91, 122, 150, 171, 201, 241, 253, 280, 307] [24, 44, 69, 92, 123, 151, 172, 202, 242, 254, 281, 308] [25, 36, 70, 93, 124, 152, 173, 203, 234, 255, 282, 309] [26, 37, 71, 94, 125, 144, 174, 204, 235, 256, 283, 310] [18, 38, 63, 95, 117, 145, 175, 205, 236, 257, 284, 311] [19, 39, 64, 96, 118, 146, 176, 206, 237, 258, 285, 312] [20, 40, 65, 97, 119, 147, 177, 198, 238, 259, 286, 313] [21, 41, 66, 98, 120, 148, 178, 199, 239, 260, 287, 314] [22, 42, 67, 90, 121, 149, 179, 200, 240, 252, 279, 306] [5, 34, 77, 100, 131, 159, 180, 210, 223, 244, 289, 316] [6, 35, 78, 101, 132, 160, 181, 211, 224, 245, 290, 317] [7, 27, 79, 102, 133, 161, 182, 212, 216, 246, 291, 318] [8, 28, 80, 103, 134, 153, 183, 213, 217, 247, 292, 319] [0, 29, 72, 104, 126, 154, 184, 214, 218, 248, 293, 320] [1, 30, 73, 105, 127, 155, 185, 215, 219, 249, 294, 321] [2, 31, 74, 106, 128, 156, 186, 207, 220, 250, 295, 322] [3, 32, 75, 107, 129, 157, 187, 208, 221, 251, 296, 323] [4, 33, 76, 99, 130, 158, 188, 209, 222, 243, 288, 315] [6, 41, 79, 86, 109, 140, 163, 189, 219, 259, 289, 298] [7, 42, 80, 87, 110, 141, 164, 190, 220, 260, 290, 299] [8, 43, 72, 88, 111, 142, 165, 191, 221, 252, 291, 300] [0, 44, 73, 89, 112, 143, 166, 192, 222, 253, 292, 301] [1, 36, 74, 81, 113, 135, 167, 193, 223, 254, 293, 302] [2, 37, 75, 82, 114, 136, 168, 194, 224, 255, 294, 303] [3, 38, 76, 83, 115, 137, 169, 195, 216, 256, 295, 304] [4, 39, 77, 84, 116, 138, 170, 196, 217, 257, 296, 305] [5, 40, 78, 85, 108, 139, 162, 197, 218, 258, 288, 297] [15, 50, 70, 95, 118, 149, 172, 198, 228, 268, 280, 307] [16, 51, 71, 96, 119, 150, 173, 199, 229, 269, 281, 308] [17, 52, 63, 97, 120, 151, 174, 200, 230, 261, 282, 309] [9, 53, 64, 98, 121, 152, 175, 201, 231, 262, 283, 310] [10, 45, 65, 90, 122, 144, 176, 202, 232, 263, 284, 311] [11, 46, 66, 91, 123, 145, 177, 203, 233, 264, 285, 312] [12, 47, 67, 92, 124, 146, 178, 204, 225, 265, 286, 313] [13, 48, 68, 93, 125, 147, 179, 205, 226, 266, 287, 314] [14, 49, 69, 94, 117, 148, 171, 206, 227, 267, 279, 306] [24, 32, 61, 104, 127, 158, 181, 207, 237, 250, 271, 316] [25, 33, 62, 105, 128, 159, 182, 208, 238, 251, 272, 317] [26, 34, 54, 106, 129, 160, 183, 209, 239, 243, 273, 318] [18, 35, 55, 107, 130, 161, 184, 210, 240, 244, 274, 319] [19, 27, 56, 99, 131, 153, 185, 211, 241, 245, 275, 320] [20, 28, 57, 100, 132, 154, 186, 212, 242, 246, 276, 321] [21, 29, 58, 101, 133, 155, 187, 213, 234, 247, 277, 322] [22, 30, 59, 102, 134, 156, 188, 214, 235, 248, 278, 323] [23, 31, 60, 103, 126, 157, 180, 215, 236, 249, 270, 315] [5, 33, 68, 106, 113, 136, 163, 190, 216, 246, 286, 316] [6, 34, 69, 107, 114, 137, 164, 191, 217, 247, 287, 317] [7, 35, 70, 99, 115, 138, 165, 192, 218, 248, 279, 318] [8, 27, 71, 100, 116, 139, 166, 193, 219, 249, 280, 319] [0, 28, 63, 101, 108, 140, 167, 194, 220, 250, 281, 320] [1, 29, 64, 102, 109, 141, 168, 195, 221, 251, 282, 321] [2, 30, 65, 103, 110, 142, 169, 196, 222, 243, 283, 322] [3, 31, 66, 104, 111, 143, 170, 197, 223, 244, 284, 323] [4, 32, 67, 105, 112, 135, 162, 189, 224, 245, 285, 315] [14, 42, 77, 97, 122, 145, 172, 199, 225, 255, 295, 307] [15, 43, 78, 98, 123, 146, 173, 200, 226, 256, 296, 308] [16, 44, 79, 90, 124, 147, 174, 201, 227, 257, 288, 309] [17, 36, 80, 91, 125, 148, 175, 202, 228, 258, 289, 310] [9, 37, 72, 92, 117, 149, 176, 203, 229, 259, 290, 311] [10, 38, 73, 93, 118, 150, 177, 204, 230, 260, 291, 312] [11, 39, 74, 94, 119, 151, 178, 205, 231, 252, 292, 313] [12, 40, 75, 95, 120, 152, 179, 206, 232, 253, 293, 314] [13, 41, 76, 96, 121, 144, 171, 198, 233, 254, 294, 306] [23, 51, 59, 88, 131, 154, 181, 208, 234, 264, 277, 298] [24, 52, 60, 89, 132, 155, 182, 209, 235, 265, 278, 299] [25, 53, 61, 81, 133, 156, 183, 210, 236, 266, 270, 300] [26, 45, 62, 82, 134, 157, 184, 211, 237, 267, 271, 301] [18, 46, 54, 83, 126, 158, 185, 212, 238, 268, 272, 302] [19, 47, 55, 84, 127, 159, 186, 213, 239, 269, 273, 303] [20, 48, 56, 85, 128, 160, 187, 214, 240, 261, 274, 304] [21, 49, 57, 86, 129, 161, 188, 215, 241, 262, 275, 305] [22, 50, 58, 87, 130, 153, 180, 207, 242, 263, 276, 297]
H_Z (81 checks, sparse supports)
[0, 35, 62, 107, 128, 141, 184, 192, 220, 251, 274, 317] [1, 27, 54, 99, 129, 142, 185, 193, 221, 243, 275, 318] [2, 28, 55, 100, 130, 143, 186, 194, 222, 244, 276, 319] [3, 29, 56, 101, 131, 135, 187, 195, 223, 245, 277, 320] [4, 30, 57, 102, 132, 136, 188, 196, 224, 246, 278, 321] [5, 31, 58, 103, 133, 137, 180, 197, 216, 247, 270, 322] [6, 32, 59, 104, 134, 138, 181, 189, 217, 248, 271, 323] [7, 33, 60, 105, 126, 139, 182, 190, 218, 249, 272, 315] [8, 34, 61, 106, 127, 140, 183, 191, 219, 250, 273, 316] [9, 44, 71, 98, 110, 150, 166, 201, 229, 260, 283, 308] [10, 36, 63, 90, 111, 151, 167, 202, 230, 252, 284, 309] [11, 37, 64, 91, 112, 152, 168, 203, 231, 253, 285, 310] [12, 38, 65, 92, 113, 144, 169, 204, 232, 254, 286, 311] [13, 39, 66, 93, 114, 145, 170, 205, 233, 255, 287, 312] [14, 40, 67, 94, 115, 146, 162, 206, 225, 256, 279, 313] [15, 41, 68, 95, 116, 147, 163, 198, 226, 257, 280, 314] [16, 42, 69, 96, 108, 148, 164, 199, 227, 258, 281, 306] [17, 43, 70, 97, 109, 149, 165, 200, 228, 259, 282, 307] [18, 53, 80, 89, 119, 159, 175, 210, 238, 269, 292, 299] [19, 45, 72, 81, 120, 160, 176, 211, 239, 261, 293, 300] [20, 46, 73, 82, 121, 161, 177, 212, 240, 262, 294, 301] [21, 47, 74, 83, 122, 153, 178, 213, 241, 263, 295, 302] [22, 48, 75, 84, 123, 154, 179, 214, 242, 264, 296, 303] [23, 49, 76, 85, 124, 155, 171, 215, 234, 265, 288, 304] [24, 50, 77, 86, 125, 156, 172, 207, 235, 266, 289, 305] [25, 51, 78, 87, 117, 157, 173, 208, 236, 267, 290, 297] [26, 52, 79, 88, 118, 158, 174, 209, 237, 268, 291, 298] [6, 27, 62, 89, 134, 155, 182, 211, 219, 247, 278, 301] [7, 28, 54, 81, 126, 156, 183, 212, 220, 248, 270, 302] [8, 29, 55, 82, 127, 157, 184, 213, 221, 249, 271, 303] [0, 30, 56, 83, 128, 158, 185, 214, 222, 250, 272, 304] [1, 31, 57, 84, 129, 159, 186, 215, 223, 251, 273, 305] [2, 32, 58, 85, 130, 160, 187, 207, 224, 243, 274, 297] [3, 33, 59, 86, 131, 161, 188, 208, 216, 244, 275, 298] [4, 34, 60, 87, 132, 153, 180, 209, 217, 245, 276, 299] [5, 35, 61, 88, 133, 154, 181, 210, 218, 246, 277, 300] [15, 36, 71, 98, 125, 137, 173, 193, 228, 256, 287, 310] [16, 37, 63, 90, 117, 138, 174, 194, 229, 257, 279, 311] [17, 38, 64, 91, 118, 139, 175, 195, 230, 258, 280, 312] [9, 39, 65, 92, 119, 140, 176, 196, 231, 259, 281, 313] [10, 40, 66, 93, 120, 141, 177, 197, 232, 260, 282, 314] [11, 41, 67, 94, 121, 142, 178, 189, 233, 252, 283, 306] [12, 42, 68, 95, 122, 143, 179, 190, 225, 253, 284, 307] [13, 43, 69, 96, 123, 135, 171, 191, 226, 254, 285, 308] [14, 44, 70, 97, 124, 136, 172, 192, 227, 255, 286, 309] [24, 45, 80, 107, 116, 146, 164, 202, 237, 265, 296, 319] [25, 46, 72, 99, 108, 147, 165, 203, 238, 266, 288, 320] [26, 47, 73, 100, 109, 148, 166, 204, 239, 267, 289, 321] [18, 48, 74, 101, 110, 149, 167, 205, 240, 268, 290, 322] [19, 49, 75, 102, 111, 150, 168, 206, 241, 269, 291, 323] [20, 50, 76, 103, 112, 151, 169, 198, 242, 261, 292, 315] [21, 51, 77, 104, 113, 152, 170, 199, 234, 262, 293, 316] [22, 52, 78, 105, 114, 144, 162, 200, 235, 263, 294, 317] [23, 53, 79, 106, 115, 145, 163, 201, 236, 264, 295, 318] [20, 33, 54, 89, 116, 161, 166, 209, 238, 246, 274, 305] [21, 34, 55, 81, 108, 153, 167, 210, 239, 247, 275, 297] [22, 35, 56, 82, 109, 154, 168, 211, 240, 248, 276, 298] [23, 27, 57, 83, 110, 155, 169, 212, 241, 249, 277, 299] [24, 28, 58, 84, 111, 156, 170, 213, 242, 250, 278, 300] [25, 29, 59, 85, 112, 157, 162, 214, 234, 251, 270, 301] [26, 30, 60, 86, 113, 158, 163, 215, 235, 243, 271, 302] [18, 31, 61, 87, 114, 159, 164, 207, 236, 244, 272, 303] [19, 32, 62, 88, 115, 160, 165, 208, 237, 245, 273, 304] [2, 42, 63, 98, 125, 152, 175, 200, 220, 255, 283, 314] [3, 43, 64, 90, 117, 144, 176, 201, 221, 256, 284, 306] [4, 44, 65, 91, 118, 145, 177, 202, 222, 257, 285, 307] [5, 36, 66, 92, 119, 146, 178, 203, 223, 258, 286, 308] [6, 37, 67, 93, 120, 147, 179, 204, 224, 259, 287, 309] [7, 38, 68, 94, 121, 148, 171, 205, 216, 260, 279, 310] [8, 39, 69, 95, 122, 149, 172, 206, 217, 252, 280, 311] [0, 40, 70, 96, 123, 150, 173, 198, 218, 253, 281, 312] [1, 41, 71, 97, 124, 151, 174, 199, 219, 254, 282, 313] [11, 51, 72, 107, 134, 143, 184, 191, 229, 264, 292, 323] [12, 52, 73, 99, 126, 135, 185, 192, 230, 265, 293, 315] [13, 53, 74, 100, 127, 136, 186, 193, 231, 266, 294, 316] [14, 45, 75, 101, 128, 137, 187, 194, 232, 267, 295, 317] [15, 46, 76, 102, 129, 138, 188, 195, 233, 268, 296, 318] [16, 47, 77, 103, 130, 139, 180, 196, 225, 269, 288, 319] [17, 48, 78, 104, 131, 140, 181, 197, 226, 261, 289, 320] [9, 49, 79, 105, 132, 141, 182, 189, 227, 262, 290, 321] [10, 50, 80, 106, 133, 142, 183, 190, 228, 263, 291, 322]
Code ID 324-166-6 · download JSON · raw on GitHub