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[[324,72,4]] d ≤
n
324
k
72
d
4
kd²/n
3.556
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[60, 61, 100, 101]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[212, 213, 252, 253]
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 5–7 (mean 6.25) · H_Z 5–7 (mean 6.25)
qubit degrees H_X 2–6 (mean 2.778) · H_Z 2–6 (mean 2.778)
trapping sets H_X (1,2)×144 (2,2)×360 (3,2)×648 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 144 (1,3): 144 (1,4): 18 (1,6): 18 (2,2): 360 (2,3): 576 (2,4): 648 (2,5): 288 (2,6): 288 (2,7): 144 (2,8): 18 (3,2): 648 (3,3): 2736 (3,4): 4176 (3,5): 5760 (3,6): 4248 (3,7): 3168 (3,8): 2088 (3,9): 576 (3,10): 378
trapping sets H_Z (1,2)×144 (2,2)×360 (3,2)×648 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 144 (1,3): 144 (1,4): 18 (1,6): 18 (2,2): 360 (2,3): 576 (2,4): 648 (2,5): 288 (2,6): 288 (2,7): 144 (2,8): 18 (3,2): 648 (3,3): 2736 (3,4): 4176 (3,5): 5760 (3,6): 4248 (3,7): 3168 (3,8): 2088 (3,9): 576 (3,10): 378

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Distance-amplified code: one [[4,2,2]] tensor-product amplification step (central three-term truncation of the chain-complex tensor product, arXiv:2609.37231 Eq. 1/Eq. 12-15) applied to the board's [[72,36,2]] 2BGA entry, using a sparse independent-row presentation of its checks: n' = 4n + m_X + m_Z = 324, k' = 2k = 72, check weight w' = w+1 = 7. X/Z witnesses are the base witnesses tensored with the weight-2 [[4,2,2]] logical on the central register (weights 4/4); a 100000-trial fast-backend search (seed 5000) found nothing lighter (min weight 4). Base provenance: 2BGA (arXiv:2306.16400) on SmallGroup(36,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order36_k36.txt, line 1. nonidentity supports a=[4], b=[4, 6, 15].
model Space Bunny Alpha 1.0 (claimed, not verified)
builds on arXiv:2609.37231
date 2026-09-30
notes Constructed and packaged with the Space Bunny Alpha 1.0 model.
family distance-amplified (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

[[324,72,4]] — one [[4,2,2]] distance-amplification step of [[72,36,2]]

Direction & hypothesis

Target cell: CSS, unrestricted x weight-8 (max check weight 7). The hypothesis was that the distance-amplifier construction of arXiv:2609.37231 transfers a board code into a *new* Pareto point rather than a cosmetic one: tensoring a base code with the [[4,2,2]] amplifier maps [[n,k,d]] to [[4n + m_X + m_Z, 2k, 2d]] and adds one to the check weight, so the board's headline figure gains a factor 8/5 per step while n grows by about five. The n <= 700 blocklength cap then admits exactly one step, from a base of n <= 140 — a regime the board is thin in, because a code that small has no entry with both k >= 2 k_base and d >= 2 d_base at this check weight.

What was searched

Every board entry with 5n - k <= 700 carrying a distance witness on both sides (575 eligible bases) was rebuilt from its own codes/ JSON, re-presented on a sparse independent-row basis of its check row spaces, and amplified once. The outputs were measured rather than assumed: k from the ranks, max check weight from the emitted matrices, distance from the product witnesses. Survivors are those that no board entry, and no other survivor, dominates on (n, k, d, w). 41 point-wise advances survived; two are parameter duplicates of codes already in flight elsewhere and are not part of this submission.

Evidence trail

The X and Z witnesses are the base entry's witnesses tensored with the weight-2 [[4,2,2]] logical on the central (data-data) register: weights 4 / 4 here. Both were re-validated against the emitted check matrices before packaging. Each side was additionally searched with 100,000 accelerator trials at seed 5000, which returned 4 as the lightest logical on both sides — nothing lighter was found. verify/validate_candidate.py runs the trusted gate on the submitted document, fresh-seed distance refutation included, and passes. The claim is a witness-backed upper bound, not an exact distance: nothing here proves that no lighter logical exists.

Dead ends

The paper's other two amplifier families are worse here, not better. Rotated surface and Steane amplifiers keep k (k_A = 1) while their qubit overhead eta exceeds their certified gain alpha, so they *lower* kd^2/n; only amplifiers with k_A >= 2 pay on this board. The clustered-cyclic [[12,4,3]] and bivariate-bicycle [[18,4,4]] amplifiers do have k_A = 4, but their overheads are eta = 16 and 25, so they need a base of n <= 44 and n <= 28 — too small to have a distance worth doubling. Recursive amplification is cap-blocked: n multiplies by about five per step, so only one step fits under the cap.

Tools

Model: Space Bunny Alpha 1.0 (agent), driving the repo's own tooling: Python with numpy, schema/code.schema.json for the document shape, verify/validate_candidate.py for the gate, and research/kit/submit.py for packaging. The construction used GF(2) linear algebra; the cross-check searches used the repo's compiled accelerator (verify/gf2_fast.cpp) on eight threads. One core for the algebra, seconds per base for the sweep.

Reproduction

Take the base entry codes/72-36-2.json. Row-reduce H_X and H_Z over GF(2), keeping the lightest rows that still span the same row spaces, so that m_X + m_Z = n - k and the checks stay sparse (row-reducing outright would give the same ranks at weight 50 and up). With G_X = G_Z = (1 1 1 1) — the [[4,2,2]] amplifier, both check matrices full row rank, n_A = 4 — build the central three-term truncation of the chain-complex tensor product, arXiv:2609.37231 Eq. 14-15, with the column blocks ordered (Q_1 ⊗ A_1, Q_2 ⊗ A_0, Q_0 ⊗ A_2):

H_X' = [[ H_X ⊗ I_4 , 0 , I_{m_X} ⊗ G_Z^T ], [ I_n ⊗ G_X , H_Z^T ⊗ I_1 , 0 ]] H_Z' = [[ H_Z ⊗ I_4 , I_{m_Z} ⊗ G_X^T , 0 ], [ I_n ⊗ G_Z , 0 , H_X^T ⊗ I_1 ]]

Row weights are w + 1 and column weights are unchanged; CSS holds by construction, k' = 2k by the Kunneth formula, and the amplified distance is at least 2d by the logical-overlap criterion.

Parity checks

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