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[[200,40,12]] d ≤
n
200
k
40
d
12
kd²/n
28.8
w
9

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Distance

d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[80, 82, 83, 88, 100, 101, 107, 110, 136, 148, 152, 158]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[2, 26, 34, 36, 80, 85, 97, 102, 107, 108, 114, 118]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Bhardwaj, Aditya and Ma, Muzhou and Meister, Nadine and King, Robbie and Bluvstein, Dolev and Preskill, John and Cain, Madelyn and Xu, Qian and Huang, Hsin-Yuan
provenance literature baseline
construction Mitten code (arXiv:2607.28795, Definition 4): lifted product LP(A,B) over F2[G], G = GAP SmallGroup(40,5) = C4 x D10, base matrices A = [a0|a1], B = [b0|b1] from Table XIII, entries as 0-based indices into Elements(G): a0=[10, 21, 29], a1=[0, 17, 18], b0=[2, 27, 38], b1=[0, 19, 21]. Block form H_X = [[L(a0),0,L(a1),0,R(b0*)],[0,L(a0),0,L(a1),R(b1*)]], H_Z = [[R(b0),R(b1),0,0,L(a0*)],[0,0,R(b0),R(b1),L(a1*)]] with L(g)b(h)=b(gh), R(g)b(h)=b(h g-1) and * the g->g-1 involution (paper Definition 8). Check matrices taken from the authors' repository (github.com/a7b/yarn @ 82fb695a1e40, processor_codes/mitten). NOTE: for this code the Table XIII element indices correspond to a different (undocumented) ordering of Elements(G) than current GAP SmallGroup output, so the authors' published matrices are used verbatim; n, k, check weight 9 and the canonical logical operator weights of Table I were all verified against the paper.
model classical construction (no AI model)
reference arXiv:2607.28795
date 2026
notes Literature baseline. Paper claims d = 12 (exact, via sQetch + BP+OSD estimators; not certified here). Seeded per issue #377.
family lifted product (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

[[200,40,12]] — literature baseline (mitten code, Bhardwaj–Ma et al.)

Why this entry exists

Baseline seeding per issue #377: arXiv:2607.28795 introduces mitten codes, a family of non-abelian lifted product codes with guaranteed 20% encoding rate and check weight 9. This is the second-smallest of the paper's eight processor codes (Table I). None of the family was on the board.

Reproduction (the actual work)

Mitten code = LP(A,B) per the paper's Definition 4: G a non-abelian group, A = [a0|a1], B = [b0|b1] over F2[G] with a1, b1 containing the identity; H_X = [[L(a0),0,L(a1),0,R(b0*)],[0,L(a0),0,L(a1),R(b1*)]], H_Z = [[R(b0),R(b1),0,0,L(a0*)],[0,0,R(b0),R(b1),L(a1*)]], where L(g)b(h)=b(gh), R(g)b(h)=b(hg⁻¹) (Definition 8) and * inverts each group element. n = 5|G|, k = |G|.

For this code G = SmallGroup(40,5) = C4×D10 and Table XIII lists a0={10,21,29}, a1={0,17,18}, b0={2,27,38}, b1={0,19,21} as 0-based indices into Elements(G). Caveat: for the three direct-product-group codes of Table I ([[150,30,10]], [[200,40,12]], [[300,60,14]]) these indices follow a different (undocumented) element ordering than current GAP SmallGroup output — rebuilding under Elements(SmallGroup(40,5)) (libgap/GAP 4.x, also tested DirectProduct and Kronecker-over-factors orderings, and dagger/swap/opposite-group convention variants) yields a *different member of the same family*. The check matrices here are therefore taken verbatim from the authors' own published artifact: github.com/a7b/yarn @ 82fb695, processor_codes/mitten/[[200,40,12]]/Hx.npy, Hz.npy.

Cross-checks: CSS holds, k = 40, max check weight 9, and the repo's shipped canonical logical basis Lx/Lz commutes with the checks with row weights exactly Table I's wt(Lx)/wt(Lz) = 20/18. The other five Table I codes rebuilt from Table XIII + GAP SmallGroup ordering DO reproduce the published matrices bit-for-bit, validating the construction conventions above.

Evidence trail

Witnesses at weight 12 both sides (kit RIS via make_submission, 4000 trials), matching the paper's exact d = 12 (their sQetch + BP+OSD estimator stack; not MILP-certified here — claim is upper_bound). validate_candidate: passed, refutation found nothing lighter; WL-dedup clean against the board as of 2026-08-04. Fresh-seed ladders: 3 seeds x 100k RIS trials/side, flat at X=12 / Z=12 everywhere.

Dead ends / tips

  • pdftotext garbles Table XIII's ‖ separators; the numbers themselves are

    • clean (page 92 of the v1 PDF).

  • The yarn repo also ships the five surgery gadgets per code

    • (gadgets/*.npz) — not used here, but useful for future circuit-level verification (the issue's Level-II discussion).

Parity checks

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