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[[300,60,14]] d ≤
n
300
k
60
d
14
kd²/n
39.2
w
9

Share this result

Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 4, 19, 22, 23, 26, 40, 50, 62, 69, 76, 86, 114, 115]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[11, 32, 33, 36, 49, 56, 120, 134, 139, 140, 142, 145, 170, 175]
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(60,11) = C10 x S3, base matrices A = [a0|a1], B = [b0|b1] from Table XIII, entries as 0-based indices into Elements(G): a0=[38, 51, 54], a1=[0, 6, 45], b0=[25, 33, 48], b1=[0, 16, 58]. 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 = 14 (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

[[300,60,14]] — 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 third 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(60,11) = C10×S3 and Table XIII lists a0={38,51,54}, a1={0,6,45}, b0={25,33,48}, b1={0,16,58} 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(60,11)) (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/[[300,60,14]]/Hx.npy, Hz.npy.

Cross-checks: CSS holds, k = 60, 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) = 22/22. 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 14 both sides (kit RIS via make_submission, 4000 trials), matching the paper's exact d = 14 (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 60k RIS trials/side, flat at X=14 / Z=14 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 120 · Z-checks 120
H_X (120 checks, sparse supports)
[3, 13, 58, 129, 166, 172, 244, 255, 297] [5, 12, 56, 131, 164, 170, 245, 254, 296] [1, 15, 59, 127, 167, 173, 241, 257, 299] [4, 14, 54, 130, 162, 168, 240, 256, 298] [0, 17, 57, 126, 165, 171, 243, 252, 294] [2, 16, 55, 128, 163, 169, 242, 253, 295] [4, 9, 19, 135, 172, 178, 243, 250, 261] [2, 11, 18, 137, 170, 176, 242, 251, 260] [5, 7, 21, 133, 173, 179, 245, 247, 263] [0, 10, 20, 136, 168, 174, 244, 246, 262] [3, 6, 23, 132, 171, 177, 240, 249, 258] [1, 8, 22, 134, 169, 175, 241, 248, 259] [10, 15, 25, 124, 141, 178, 249, 256, 267] [8, 17, 24, 122, 143, 176, 248, 257, 266] [11, 13, 27, 125, 139, 179, 251, 253, 269] [6, 16, 26, 120, 142, 174, 250, 252, 268] [9, 12, 29, 123, 138, 177, 246, 255, 264] [7, 14, 28, 121, 140, 175, 247, 254, 265] [16, 21, 31, 124, 130, 147, 255, 262, 273] [14, 23, 30, 122, 128, 149, 254, 263, 272] [17, 19, 33, 125, 131, 145, 257, 259, 275] [12, 22, 32, 120, 126, 148, 256, 258, 274] [15, 18, 35, 123, 129, 144, 252, 261, 270] [13, 20, 34, 121, 127, 146, 253, 260, 271] [22, 27, 37, 130, 136, 153, 261, 268, 279] [20, 29, 36, 128, 134, 155, 260, 269, 278] [23, 25, 39, 131, 137, 151, 263, 265, 281] [18, 28, 38, 126, 132, 154, 262, 264, 280] [21, 24, 41, 129, 135, 150, 258, 267, 276] [19, 26, 40, 127, 133, 152, 259, 266, 277] [28, 33, 43, 136, 142, 159, 267, 274, 285] [26, 35, 42, 134, 140, 161, 266, 275, 284] [29, 31, 45, 137, 143, 157, 269, 271, 287] [24, 34, 44, 132, 138, 160, 268, 270, 286] [27, 30, 47, 135, 141, 156, 264, 273, 282] [25, 32, 46, 133, 139, 158, 265, 272, 283] [34, 39, 49, 142, 148, 165, 273, 280, 291] [32, 41, 48, 140, 146, 167, 272, 281, 290] [35, 37, 51, 143, 149, 163, 275, 277, 293] [30, 40, 50, 138, 144, 166, 274, 276, 292] [33, 36, 53, 141, 147, 162, 270, 279, 288] [31, 38, 52, 139, 145, 164, 271, 278, 289] [40, 45, 55, 148, 154, 171, 279, 286, 297] [38, 47, 54, 146, 152, 173, 278, 287, 296] [41, 43, 57, 149, 155, 169, 281, 283, 299] [36, 46, 56, 144, 150, 172, 280, 282, 298] [39, 42, 59, 147, 153, 168, 276, 285, 294] [37, 44, 58, 145, 151, 170, 277, 284, 295] [1, 46, 51, 154, 160, 177, 243, 285, 292] [0, 44, 53, 152, 158, 179, 242, 284, 293] [3, 47, 49, 155, 161, 175, 245, 287, 289] [2, 42, 52, 150, 156, 178, 244, 286, 288] [5, 45, 48, 153, 159, 174, 240, 282, 291] [4, 43, 50, 151, 157, 176, 241, 283, 290] [7, 52, 57, 123, 160, 166, 249, 291, 298] [6, 50, 59, 125, 158, 164, 248, 290, 299] [9, 53, 55, 121, 161, 167, 251, 293, 295] [8, 48, 58, 124, 156, 162, 250, 292, 294] [11, 51, 54, 120, 159, 165, 246, 288, 297] [10, 49, 56, 122, 157, 163, 247, 289, 296] [63, 73, 118, 189, 226, 232, 266, 270, 292] [65, 72, 116, 191, 224, 230, 267, 271, 293] [61, 75, 119, 187, 227, 233, 264, 272, 289] [64, 74, 114, 190, 222, 228, 265, 273, 288] [60, 77, 117, 186, 225, 231, 269, 274, 291] [62, 76, 115, 188, 223, 229, 268, 275, 290] [64, 69, 79, 195, 232, 238, 272, 276, 298] [62, 71, 78, 197, 230, 236, 273, 277, 299] [65, 67, 81, 193, 233, 239, 270, 278, 295] [60, 70, 80, 196, 228, 234, 271, 279, 294] [63, 66, 83, 192, 231, 237, 275, 280, 297] [61, 68, 82, 194, 229, 235, 274, 281, 296] [70, 75, 85, 184, 201, 238, 244, 278, 282] [68, 77, 84, 182, 203, 236, 245, 279, 283] [71, 73, 87, 185, 199, 239, 241, 276, 284] [66, 76, 86, 180, 202, 234, 240, 277, 285] [69, 72, 89, 183, 198, 237, 243, 281, 286] [67, 74, 88, 181, 200, 235, 242, 280, 287] [76, 81, 91, 184, 190, 207, 250, 284, 288] [74, 83, 90, 182, 188, 209, 251, 285, 289] [77, 79, 93, 185, 191, 205, 247, 282, 290] [72, 82, 92, 180, 186, 208, 246, 283, 291] [75, 78, 95, 183, 189, 204, 249, 287, 292] [73, 80, 94, 181, 187, 206, 248, 286, 293] [82, 87, 97, 190, 196, 213, 256, 290, 294] [80, 89, 96, 188, 194, 215, 257, 291, 295] [83, 85, 99, 191, 197, 211, 253, 288, 296] [78, 88, 98, 186, 192, 214, 252, 289, 297] [81, 84, 101, 189, 195, 210, 255, 293, 298] [79, 86, 100, 187, 193, 212, 254, 292, 299] [88, 93, 103, 196, 202, 219, 240, 262, 296] [86, 95, 102, 194, 200, 221, 241, 263, 297] [89, 91, 105, 197, 203, 217, 242, 259, 294] [84, 94, 104, 192, 198, 220, 243, 258, 295] [87, 90, 107, 195, 201, 216, 244, 261, 299] [85, 92, 106, 193, 199, 218, 245, 260, 298] [94, 99, 109, 202, 208, 225, 242, 246, 268] [92, 101, 108, 200, 206, 227, 243, 247, 269] [95, 97, 111, 203, 209, 223, 240, 248, 265] [90, 100, 110, 198, 204, 226, 241, 249, 264] [93, 96, 113, 201, 207, 222, 245, 250, 267] [91, 98, 112, 199, 205, 224, 244, 251, 266] [100, 105, 115, 208, 214, 231, 248, 252, 274] [98, 107, 114, 206, 212, 233, 249, 253, 275] [101, 103, 117, 209, 215, 229, 246, 254, 271] [96, 106, 116, 204, 210, 232, 247, 255, 270] [99, 102, 119, 207, 213, 228, 251, 256, 273] [97, 104, 118, 205, 211, 230, 250, 257, 272] [61, 106, 111, 214, 220, 237, 254, 258, 280] [60, 104, 113, 212, 218, 239, 255, 259, 281] [63, 107, 109, 215, 221, 235, 252, 260, 277] [62, 102, 112, 210, 216, 238, 253, 261, 276] [65, 105, 108, 213, 219, 234, 257, 262, 279] [64, 103, 110, 211, 217, 236, 256, 263, 278] [67, 112, 117, 183, 220, 226, 260, 264, 286] [66, 110, 119, 185, 218, 224, 261, 265, 287] [69, 113, 115, 181, 221, 227, 258, 266, 283] [68, 108, 118, 184, 216, 222, 259, 267, 282] [71, 111, 114, 180, 219, 225, 263, 268, 285] [70, 109, 116, 182, 217, 223, 262, 269, 284]
H_Z (120 checks, sparse supports)
[3, 10, 52, 75, 90, 98, 244, 249, 289] [2, 11, 53, 74, 91, 99, 242, 251, 288] [5, 7, 49, 77, 92, 96, 245, 247, 291] [4, 6, 48, 76, 93, 97, 240, 250, 290] [0, 9, 51, 72, 94, 101, 243, 246, 293] [1, 8, 50, 73, 95, 100, 241, 248, 292] [9, 16, 58, 81, 96, 104, 250, 255, 295] [8, 17, 59, 80, 97, 105, 248, 257, 294] [11, 13, 55, 83, 98, 102, 251, 253, 297] [10, 12, 54, 82, 99, 103, 246, 256, 296] [6, 15, 57, 78, 100, 107, 249, 252, 299] [7, 14, 56, 79, 101, 106, 247, 254, 298] [4, 15, 22, 87, 102, 110, 241, 256, 261] [5, 14, 23, 86, 103, 111, 240, 254, 263] [1, 17, 19, 89, 104, 108, 243, 257, 259] [0, 16, 18, 88, 105, 109, 242, 252, 262] [3, 12, 21, 84, 106, 113, 245, 255, 258] [2, 13, 20, 85, 107, 112, 244, 253, 260] [10, 21, 28, 93, 108, 116, 247, 262, 267] [11, 20, 29, 92, 109, 117, 246, 260, 269] [7, 23, 25, 95, 110, 114, 249, 263, 265] [6, 22, 24, 94, 111, 115, 248, 258, 268] [9, 18, 27, 90, 112, 119, 251, 261, 264] [8, 19, 26, 91, 113, 118, 250, 259, 266] [16, 27, 34, 62, 99, 114, 253, 268, 273] [17, 26, 35, 63, 98, 115, 252, 266, 275] [13, 29, 31, 60, 101, 116, 255, 269, 271] [12, 28, 30, 61, 100, 117, 254, 264, 274] [15, 24, 33, 65, 96, 118, 257, 267, 270] [14, 25, 32, 64, 97, 119, 256, 265, 272] [22, 33, 40, 60, 68, 105, 259, 274, 279] [23, 32, 41, 61, 69, 104, 258, 272, 281] [19, 35, 37, 62, 66, 107, 261, 275, 277] [18, 34, 36, 63, 67, 106, 260, 270, 280] [21, 30, 39, 64, 71, 102, 263, 273, 276] [20, 31, 38, 65, 70, 103, 262, 271, 278] [28, 39, 46, 66, 74, 111, 265, 280, 285] [29, 38, 47, 67, 75, 110, 264, 278, 287] [25, 41, 43, 68, 72, 113, 267, 281, 283] [24, 40, 42, 69, 73, 112, 266, 276, 286] [27, 36, 45, 70, 77, 108, 269, 279, 282] [26, 37, 44, 71, 76, 109, 268, 277, 284] [34, 45, 52, 72, 80, 117, 271, 286, 291] [35, 44, 53, 73, 81, 116, 270, 284, 293] [31, 47, 49, 74, 78, 119, 273, 287, 289] [30, 46, 48, 75, 79, 118, 272, 282, 292] [33, 42, 51, 76, 83, 114, 275, 285, 288] [32, 43, 50, 77, 82, 115, 274, 283, 290] [40, 51, 58, 63, 78, 86, 277, 292, 297] [41, 50, 59, 62, 79, 87, 276, 290, 299] [37, 53, 55, 65, 80, 84, 279, 293, 295] [36, 52, 54, 64, 81, 85, 278, 288, 298] [39, 48, 57, 60, 82, 89, 281, 291, 294] [38, 49, 56, 61, 83, 88, 280, 289, 296] [4, 46, 57, 69, 84, 92, 243, 283, 298] [5, 47, 56, 68, 85, 93, 245, 282, 296] [1, 43, 59, 71, 86, 90, 241, 285, 299] [0, 42, 58, 70, 87, 91, 244, 284, 294] [3, 45, 54, 66, 88, 95, 240, 287, 297] [2, 44, 55, 67, 89, 94, 242, 286, 295] [123, 130, 172, 195, 210, 218, 255, 261, 298] [122, 131, 173, 194, 211, 219, 257, 263, 296] [125, 127, 169, 197, 212, 216, 253, 259, 299] [124, 126, 168, 196, 213, 217, 256, 262, 294] [120, 129, 171, 192, 214, 221, 252, 258, 297] [121, 128, 170, 193, 215, 220, 254, 260, 295] [129, 136, 178, 201, 216, 224, 244, 261, 267] [128, 137, 179, 200, 217, 225, 242, 263, 269] [131, 133, 175, 203, 218, 222, 245, 259, 265] [130, 132, 174, 202, 219, 223, 240, 262, 268] [126, 135, 177, 198, 220, 227, 243, 258, 264] [127, 134, 176, 199, 221, 226, 241, 260, 266] [124, 135, 142, 207, 222, 230, 250, 267, 273] [125, 134, 143, 206, 223, 231, 248, 269, 275] [121, 137, 139, 209, 224, 228, 251, 265, 271] [120, 136, 138, 208, 225, 229, 246, 268, 274] [123, 132, 141, 204, 226, 233, 249, 264, 270] [122, 133, 140, 205, 227, 232, 247, 266, 272] [130, 141, 148, 213, 228, 236, 256, 273, 279] [131, 140, 149, 212, 229, 237, 254, 275, 281] [127, 143, 145, 215, 230, 234, 257, 271, 277] [126, 142, 144, 214, 231, 235, 252, 274, 280] [129, 138, 147, 210, 232, 239, 255, 270, 276] [128, 139, 146, 211, 233, 238, 253, 272, 278] [136, 147, 154, 182, 219, 234, 262, 279, 285] [137, 146, 155, 183, 218, 235, 260, 281, 287] [133, 149, 151, 180, 221, 236, 263, 277, 283] [132, 148, 150, 181, 220, 237, 258, 280, 286] [135, 144, 153, 185, 216, 238, 261, 276, 282] [134, 145, 152, 184, 217, 239, 259, 278, 284] [142, 153, 160, 180, 188, 225, 268, 285, 291] [143, 152, 161, 181, 189, 224, 266, 287, 293] [139, 155, 157, 182, 186, 227, 269, 283, 289] [138, 154, 156, 183, 187, 226, 264, 286, 292] [141, 150, 159, 184, 191, 222, 267, 282, 288] [140, 151, 158, 185, 190, 223, 265, 284, 290] [148, 159, 166, 186, 194, 231, 274, 291, 297] [149, 158, 167, 187, 195, 230, 272, 293, 299] [145, 161, 163, 188, 192, 233, 275, 289, 295] [144, 160, 162, 189, 193, 232, 270, 292, 298] [147, 156, 165, 190, 197, 228, 273, 288, 294] [146, 157, 164, 191, 196, 229, 271, 290, 296] [154, 165, 172, 192, 200, 237, 243, 280, 297] [155, 164, 173, 193, 201, 236, 245, 278, 299] [151, 167, 169, 194, 198, 239, 241, 281, 295] [150, 166, 168, 195, 199, 238, 244, 276, 298] [153, 162, 171, 196, 203, 234, 240, 279, 294] [152, 163, 170, 197, 202, 235, 242, 277, 296] [160, 171, 178, 183, 198, 206, 243, 249, 286] [161, 170, 179, 182, 199, 207, 245, 251, 284] [157, 173, 175, 185, 200, 204, 241, 247, 287] [156, 172, 174, 184, 201, 205, 244, 250, 282] [159, 168, 177, 180, 202, 209, 240, 246, 285] [158, 169, 176, 181, 203, 208, 242, 248, 283] [124, 166, 177, 189, 204, 212, 249, 255, 292] [125, 167, 176, 188, 205, 213, 251, 257, 290] [121, 163, 179, 191, 206, 210, 247, 253, 293] [120, 162, 178, 190, 207, 211, 250, 256, 288] [123, 165, 174, 186, 208, 215, 246, 252, 291] [122, 164, 175, 187, 209, 214, 248, 254, 289]
raw submission JSON