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[[336,58,6]] d ≤
n
336
k
58
d
6
kd²/n
6.214
w
6
X/Z
1.33

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Distance

X/Z asymmetry 1.33 · d_X ≤ 8, d_Z ≤ 6 · w_X = 6, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[54, 88, 89, 169, 170, 254, 288, 299]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[144, 187, 215, 233, 279, 313]
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 8 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×336 (2,2)×1680 (3,2)×8400 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 336 (2,2): 1680 (3,2): 8400 (3,4): 2240
trapping sets H_Z (1,2)×336 (2,2)×1008 (3,2)×3024 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 336 (2,2): 1008 (3,2): 3024 (3,4): 672

Construction & provenance

authors @msilve160
provenance submitted through the challenge
novelty novelty not audited
construction Planar hyperbolic {4,6} code, literature reproduction from github.com/QEC-pages/Quantum_LDPC_Codes @ 1c95489383564e4dc2cce517de00d64d6f2c4f56, Hyperbolic_Codes_Planar.zip, parity-check matrices for N=336. Same source repository already used for codes/720-122-8.json, codes/864-146-8.json, codes/896-194-6.json, codes/900-182-8.json, codes/960-258-6.json (Seam D, fieldnotes/2026-09-18-hackathon-1155-frontier-map-and-playbook.md); this pulls a different (p,q,N) entry from the same pinned catalog (Hyperbolic_Codes.tsv) that those five did not use. CSS code, edge model.
model Claude Claude Sonnet 5 (claimed, not verified)
builds on github.com/QEC-pages/Quantum_LDPC_Codes
date 2026-09-19
family topological (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[336,58,6]] — planar hyperbolic {4,6} code

Credit

Literature reproduction, not an original construction: matrices come unmodified from github.com/QEC-pages/Quantum_LDPC_Codes @ 1c95489383564e4dc2cce517de00d64d6f2c4f56 (Hyperbolic_Codes_Planar.zip), the same pinned source @MathysRennela (using DeepSeek V4 Flash 0731) already used for the five topological-family "Seam D" codes in fieldnotes/2026-09-18-hackathon-1155-frontier-map-and-playbook.md, including codes/864-146-8.json — also a {4,6} tessellation, at N=864 rather than this submission's N=336.

Direction & hypothesis

Same direction as 80-18-5.note.md: filtering the source repo's own Hyperbolic_Codes.tsv index to N<=1000 gives 21 entries; only 5 (Seam D) were on the board. This is one of the 6 that checked out as still open. Note the board also has a {4,6} N=660 entry from this same index checked in this sweep, but that one was found dominated (not submitted); N=336 was not.

What was searched

Same sweep as 80-18-5.note.md. Independently computed k matched the file's own header-comment [[n,k,d]] ([[336,58,6]] here).

Evidence trail

  • css.verify_css: CSS commutation confirmed.
  • css.compute_k: k=58, matching the file's own header.
  • submit.make_submission witness search (8,000 RIS trials/side, numpy,

    • seed 0): found a weight-6 witness on both sides, matching the paper's claimed distance.

  • Gate: verify/validate_candidate.py returned passed: true,

    • refute.refuted: false, novelty.board_advancing: true for (weight-6, unrestricted), dominated_by: [].

  • Claim: witness-backed upper bound, not exact.

Dead ends

See 80-18-5.note.md for the full sweep. Specific to this family: the {4,6} N=660 entry from the same catalog (k=112, d=6) was found dominated by codes/540-112-8.json and was not submitted.

Tools

Same as 80-18-5.note.md.

Reproduction

git clone https://github.com/QEC-pages/Quantum_LDPC_Codes.git
cd Quantum_LDPC_Codes
unzip Hyperbolic_Codes_Planar.zip -d hyp
# HX, HZ: hyp/4_6/4_6_336.mtx and hyp/4_6/6_4_336.mtx

import sys
sys.path.insert(0, "research/kit")
from submit import make_submission, save_submission
doc = make_submission(
    HX, HZ,
    name="[[336,58,6]] planar hyperbolic {4,6} code",
    construction="Planar hyperbolic {4,6} code, literature reproduction from "
                 "github.com/QEC-pages/Quantum_LDPC_Codes @ "
                 "1c95489383564e4dc2cce517de00d64d6f2c4f56, N=336.",
    authors=["@msilve160"], family="topological",
    references=["github.com/QEC-pages/Quantum_LDPC_Codes"],
    confidence="upper_bound", trials=8000, seed=0,
)
save_submission(doc, "codes/336-58-6.json")   # only after human review + PR

Parity checks

X-checks 112 (max weight 6) · Z-checks 168 (max weight 4)
H_X (112 checks, sparse supports)
[0, 2, 3, 7, 9, 18] [0, 1, 5, 11, 13, 25] [2, 4, 6, 14, 16, 29] [1, 4, 10, 21, 23, 41] [7, 15, 17, 32, 34, 61] [3, 8, 19, 36, 38, 68] [8, 11, 24, 44, 46, 82] [5, 12, 26, 48, 50, 89] [14, 22, 28, 52, 54, 95] [6, 15, 30, 56, 58, 102] [18, 33, 35, 64, 66, 116] [9, 20, 39, 65, 72, 119] [12, 21, 40, 73, 75, 131] [10, 22, 42, 77, 79, 138] [25, 45, 47, 85, 87, 152] [13, 27, 51, 86, 93, 155] [29, 53, 55, 98, 100, 175] [16, 31, 59, 99, 106, 178] [32, 57, 60, 107, 109, 164] [17, 33, 62, 111, 113, 193] [20, 36, 67, 120, 122, 203] [19, 37, 69, 124, 126, 209] [41, 74, 76, 134, 136, 222] [23, 43, 80, 135, 142, 225] [37, 44, 81, 143, 145, 234] [24, 45, 83, 147, 149, 240] [27, 48, 88, 156, 158, 250] [26, 49, 90, 160, 162, 256] [52, 78, 94, 128, 166, 168] [28, 53, 96, 170, 172, 252] [31, 56, 101, 179, 181, 273] [30, 57, 103, 183, 185, 278] [61, 108, 110, 163, 174, 267] [34, 63, 114, 169, 192, 263] [64, 112, 115, 162, 198, 271] [35, 65, 117, 158, 199, 252] [68, 121, 123, 159, 206, 254] [38, 70, 127, 165, 207, 257] [39, 71, 128, 210, 211, 290] [49, 73, 130, 187, 213, 215] [40, 74, 132, 205, 217, 219] [43, 77, 137, 226, 228, 304] [42, 78, 139, 230, 232, 309] [82, 115, 144, 146, 197, 233] [46, 84, 110, 150, 192, 239] [85, 148, 151, 201, 232, 245] [47, 86, 153, 193, 228, 246] [89, 157, 159, 229, 253, 307] [50, 91, 163, 235, 254, 310] [51, 92, 164, 257, 258, 321] [95, 127, 167, 169, 221, 298] [54, 97, 173, 216, 263, 294] [98, 126, 171, 174, 268, 302] [55, 99, 122, 176, 205, 269] [102, 123, 180, 182, 207, 275] [58, 104, 129, 186, 210, 276] [59, 105, 187, 279, 280, 319] [69, 107, 125, 184, 189, 264] [60, 108, 190, 203, 282, 283] [63, 111, 137, 227, 284, 305] [62, 112, 194, 266, 285, 318] [116, 142, 197, 235, 281, 312] [66, 118, 201, 225, 258, 301] [119, 136, 200, 224, 280, 302] [71, 120, 202, 251, 260, 317] [67, 121, 133, 204, 217, 287] [70, 113, 124, 195, 208, 261] [72, 129, 212, 222, 259, 298] [131, 151, 186, 214, 216, 244] [75, 133, 146, 220, 239, 294] [134, 185, 218, 221, 248, 299] [76, 135, 181, 223, 240, 300] [138, 182, 227, 229, 276, 306] [79, 140, 188, 233, 279, 307] [80, 141, 234, 288, 310, 311] [90, 125, 143, 161, 194, 236] [81, 144, 237, 250, 313, 314] [84, 101, 147, 180, 274, 315] [83, 148, 196, 241, 284, 316] [106, 152, 188, 212, 244, 281] [87, 154, 178, 248, 270, 311] [100, 155, 177, 211, 247, 271] [92, 156, 189, 249, 305, 331] [88, 97, 157, 170, 251, 318] [91, 149, 160, 190, 242, 255] [93, 165, 175, 259, 267, 312] [103, 166, 184, 231, 260, 295] [94, 167, 261, 273, 322, 323] [96, 171, 264, 287, 297, 324] [105, 179, 204, 272, 286, 291] [104, 172, 183, 265, 277, 292] [109, 191, 223, 262, 301, 322] [114, 196, 206, 286, 288, 299] [118, 199, 277, 309, 326, 332] [117, 200, 215, 238, 293, 313] [209, 224, 255, 289, 300, 320] [139, 161, 213, 231, 241, 291] [130, 214, 292, 304, 328, 329] [132, 218, 243, 295, 315, 330] [141, 226, 236, 274, 303, 325] [140, 219, 230, 237, 296, 308] [145, 176, 191, 238, 270, 282] [150, 243, 253, 268, 317, 319] [154, 208, 246, 278, 289, 326] [153, 168, 247, 262, 293, 328] [177, 256, 269, 308, 320, 332] [173, 245, 266, 275, 321, 325] [195, 202, 283, 285, 327, 334] [198, 220, 290, 297, 306, 331] [242, 249, 314, 316, 327, 333] [265, 272, 323, 324, 334, 335] [296, 303, 329, 330, 333, 335]
H_Z (168 checks, sparse supports)
[0, 1, 2, 4] [2, 6, 7, 15] [0, 3, 8, 11] [1, 5, 12, 21] [4, 10, 14, 22] [7, 17, 18, 33] [3, 9, 20, 36] [11, 24, 25, 45] [5, 13, 27, 48] [14, 28, 29, 53] [6, 16, 31, 56] [15, 30, 32, 57] [9, 18, 35, 65] [8, 19, 37, 44] [21, 40, 41, 74] [10, 23, 43, 77] [13, 25, 47, 86] [12, 26, 49, 73] [22, 42, 52, 78] [16, 29, 55, 99] [32, 60, 61, 108] [17, 34, 63, 111] [33, 62, 64, 112] [36, 67, 68, 121] [19, 38, 70, 124] [20, 39, 71, 120] [23, 41, 76, 135] [44, 81, 82, 144] [24, 46, 84, 147] [45, 83, 85, 148] [48, 88, 89, 157] [26, 50, 91, 160] [27, 51, 92, 156] [52, 94, 95, 167] [28, 54, 97, 170] [53, 96, 98, 171] [56, 101, 102, 180] [30, 58, 104, 183] [31, 59, 105, 179] [57, 103, 107, 184] [34, 61, 110, 192] [64, 115, 116, 197] [35, 66, 118, 199] [65, 117, 119, 200] [38, 68, 123, 207] [37, 69, 125, 143] [39, 72, 129, 210] [73, 130, 131, 214] [40, 75, 133, 217] [74, 132, 134, 218] [77, 137, 138, 227] [42, 79, 140, 230] [43, 80, 141, 226] [46, 82, 146, 239] [85, 151, 152, 244] [47, 87, 154, 246] [86, 153, 155, 247] [50, 89, 159, 254] [49, 90, 161, 213] [51, 93, 165, 257] [78, 139, 166, 231] [54, 95, 169, 263] [98, 174, 175, 267] [55, 100, 177, 269] [99, 176, 178, 270] [58, 102, 182, 276] [59, 106, 188, 279] [92, 107, 164, 189] [60, 109, 191, 282] [91, 108, 163, 190] [111, 137, 193, 228] [62, 113, 195, 285] [63, 114, 196, 284] [90, 112, 162, 194] [66, 116, 142, 225] [72, 119, 136, 222] [120, 202, 203, 283] [67, 122, 205, 217] [121, 204, 206, 286] [124, 208, 209, 289] [69, 126, 171, 264] [70, 127, 167, 261] [71, 128, 166, 260] [75, 131, 216, 294] [134, 221, 222, 298] [76, 136, 224, 300] [135, 223, 225, 301] [79, 138, 229, 307] [80, 142, 235, 310] [141, 143, 234, 236] [81, 145, 238, 313] [140, 144, 233, 237] [101, 147, 181, 240] [83, 149, 242, 316] [84, 150, 243, 315] [139, 148, 232, 241] [87, 106, 152, 178] [93, 100, 155, 175] [156, 249, 250, 314] [88, 158, 170, 252] [157, 251, 253, 317] [160, 255, 256, 320] [94, 168, 262, 322] [96, 172, 265, 324] [97, 173, 266, 318] [179, 272, 273, 323] [180, 274, 275, 325] [183, 277, 278, 326] [103, 185, 218, 295] [104, 186, 214, 292] [105, 187, 213, 291] [125, 161, 184, 231] [109, 164, 258, 301] [163, 235, 267, 312] [110, 150, 174, 268] [192, 239, 263, 294] [113, 193, 208, 246] [114, 169, 221, 299] [162, 177, 256, 271] [115, 146, 198, 220] [188, 197, 233, 281] [172, 199, 252, 277] [117, 158, 250, 313] [118, 201, 232, 309] [187, 200, 215, 280] [122, 176, 203, 282] [206, 254, 288, 310] [123, 159, 182, 229] [207, 257, 275, 321] [126, 209, 224, 302] [127, 165, 259, 298] [210, 276, 290, 306] [128, 168, 211, 247] [129, 186, 212, 244] [130, 215, 293, 328] [132, 219, 296, 330] [133, 220, 287, 297] [226, 303, 304, 329] [227, 305, 306, 331] [230, 308, 309, 332] [145, 234, 270, 311] [149, 240, 255, 300] [151, 173, 216, 245] [153, 228, 304, 328] [154, 185, 248, 278] [253, 279, 307, 319] [205, 219, 269, 308] [181, 223, 273, 322] [189, 264, 297, 331] [190, 242, 283, 327] [191, 238, 262, 293] [249, 284, 305, 316] [202, 251, 285, 318] [194, 236, 266, 325] [195, 261, 323, 334] [196, 241, 286, 291] [198, 211, 271, 290] [212, 259, 281, 312] [201, 245, 258, 321] [268, 280, 302, 319] [243, 260, 295, 317] [204, 272, 287, 324] [289, 320, 326, 332] [237, 296, 314, 333] [274, 303, 315, 330] [248, 288, 299, 311] [265, 292, 329, 335] [327, 333, 334, 335]
Code ID 336-58-6 · download JSON · raw on GitHub