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[[342,40,32]] d ≤
n
342
k
40
d
32
kd²/n
119.766
w
14
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 32, d_Z ≤ 32 · w_X = 14, w_Z = 14 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 32 · witness weight 32 (claimed upper_bound)
witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 400 trials · survived 400 trials · 2026-10-15
witness operator (support, 32 qubits)
[17, 52, 54, 61, 100, 107, 113, 114, 145, 147, 152, 168, 169, 173, 175, 178, 198, 204, 207, 219, 228, 230, 261, 273, 279, 281, 298, 301, 308, 322, 331, 336]
d_Z 32 · witness weight 32 (claimed upper_bound)
witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 400 trials · survived 400 trials · 2026-10-15
witness operator (support, 32 qubits)
[28, 29, 32, 67, 75, 80, 97, 104, 114, 134, 151, 156, 157, 173, 176, 177, 195, 203, 212, 216, 229, 233, 237, 259, 261, 264, 277, 278, 287, 319, 320, 339]
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 14 · H_Z 14
qubit degrees H_X 5–9 (mean 7.0) · H_Z 5–9 (mean 7.0)
trapping sets H_X (1,5)×171 (2,6)×684 (3,7)×2394 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 171 (1,9): 171 (2,6): 684 (2,8): 342 (2,12): 7695 (2,14): 855 (2,16): 4446 (3,7): 2394 (3,9): 4446 (3,11): 3762 (3,13): 33858 (3,15): 184680 (3,17): 89946 (3,19): 415188 (3,21): 93537 (3,23): 130986 (3,25): 6156
trapping sets H_Z (1,5)×171 (2,6)×684 (3,7)×2394 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 171 (1,9): 171 (2,6): 684 (2,8): 342 (2,12): 7695 (2,14): 855 (2,16): 4446 (3,7): 2394 (3,9): 4446 (3,11): 3762 (3,13): 33858 (3,15): 184680 (3,17): 89946 (3,19): 415188 (3,21): 93537 (3,23): 130986 (3,25): 6156

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Cyclic generalized bicycle at m=171 = 3 x 3 x 19, with ord_2(m)=18, so x171 - 1 factors over GF(2) into 12 cyclotomic cosets, 1 of size 1, 1 of size 2, 1 of size 6, 9 of size 18. The divisor g is assembled from coset minimal polynomials to degree 20, which fixes k = 2 deg g = 40 by construction, and (a,b) was drawn from the ideal (g) by information-set decoding over sparse supports, keeping weight 5 and weight 9 in the two blocks. At least one weight is odd, so x+1 does not divide the gcd. m is composite, so the quotient fold acts at t=3, t=9, t=19 and t=57; a fold over every equal-orbit partition of the qubits returns nothing lighter than 48, clear of the claim. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 342 qubits at uniform row weight 14, with a = [67, 71, 118, 165, 169] and b = [13, 14, 15, 49, 58, 79, 97, 132, 159].
model Claude Claude Opus 5 (claimed, not verified)
date 2026-10-15
family generalized bicycle (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

--- author: "@vprusso" ---

[[342,40,32]]: cyclic generalized-bicycle code at m = 171, row weight 14

At row weight 14 or lighter, the best distance on the board with n <= 342 and k >= 40 is 22, at [[254,42,22]] and row weight 13. This entry reaches 32. Allowing any row weight the best in the same range is 24, at [[254,44,24]] and row weight 16.

Direction & hypothesis

m = 171 had never been swept for this construction, and n = 342 held three board entries, at k = 6, k = 46 and k = 76, with distances 39, 3 and 4. Nothing sat at this dimension.

For a cyclic generalized bicycle on x^m - 1 the dimension is k = 2 deg gcd(a, b, x^m - 1), so it can be set in advance: build a divisor g of chosen degree out of cyclotomic coset minimal polynomials and draw a and b as low-weight codewords of its ideal. Degree 20 at m = 171 is deg g / m of 0.117, at the bottom of the band this family produces anything in; every candidate it yielded in the same session sat between 0.117 and 0.152, and degrees above roughly 0.17 return distance 4 or 5 at any modulus.

What was searched

m = 171 = 3^2 x 19 with ord_2(171) = 18, so x^171 - 1 factors over GF(2) into 12 cyclotomic cosets: one of size 1, one of size 2, one of size 6 and nine of size 18. Reachable divisor degrees are the subset sums of those sizes, and the divisor here has degree 20, fixing k = 40.

Degrees were filtered before cores were spent: the degree must be a sum of coset sizes or no divisor exists, and the chosen exponents must not share a factor with m or the divisor sits inside x^q - 1 for a proper divisor q and the codes fall below distance 4.

The sweep ran at 20,000 information-set trials per draw. This design point produced draws across row weights 14 through 24, and they are separate board cells rather than competitors, because domination needs the dominator to be no heavier and at least as far: a lighter draw at a lower distance does not displace a heavier one at a higher distance. This is the lightest draw the point produced. The polynomial weights are 5 and 9, so (x + 1) does not divide the common gcd.

Evidence trail

deg gcd(a, b, x^171 - 1) = 20, and the same degree on each side alone, so the leak between the two single-block kernels and k is 0.

A 24-seed gate at 2,000,000 information-set trials per seed settled at 32, and a single uninterrupted run of 8,000,000 trials, matching the depth the verifier's own refutation pass reaches, read the same 32. Seven of the 24 gate seeds reached the minimum, with the spread running 32 to 38. Both the agreement between two instruments at different depths and the width of that minimum are the strongest of any candidate this construction produced in the session; elsewhere the two instruments disagreed by 2 to 8 and typically one or two seeds of 24 found the minimum.

The single-block logicals form the cyclic code of dimension 20 whose zeros are the 151 residues the divisor omits. The longest run of consecutive residues among them is 25, so the Bose-Chaudhuri-Hocquenghem bound puts the lightest single-block logical at 26 or above, and a direct search confined to one circulant block at 12 seeds of 600,000 trials independently finds nothing below 34. The quotient fold acts at t = 3, 9, 19 and 57; a fold over every equal-orbit partition of the qubits returns nothing lighter than 48, so none of the four folds binds.

The claim is a witness-backed upper bound, not certified. The X witness of weight 32 is the operator the gate found; the Z witness is that operator carried across by the construction's own symmetry, since H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] means swapping the blocks and negating the index inside each maps one side's logicals onto the other's. It was validated against the opposite check matrix and the logical basis before being written, and both witnesses commute with the opposite check matrix and lie outside its row space.

Dead ends

Three heavier draws at this same dimension were measured. Row weight 16 reached 34 and row weight 20 reached 36, both of which are separate cells from this one and neither of which dominates it. Row weight 22 was dominated by the weight-20 draw outright.

Lighter is not automatically better in this family, and this dimension shows why. The board's bar at [[342,40]] is 24 at row weight 20, 24 at weight 16 and 22 at weight 14, so dropping two or four weight steps buys almost nothing, while the distance falls from 36 to 34 to 32. The lighter draws survive here only because the bar happens to fall by 2 at weight 14; at [[342,42]] the bar is 24 at every weight from 20 down to 16 and a lighter draw there is pure loss.

Tools

Claude Opus 5 driving a Claude Code session. Distance readings, witness validation and the single-block searches come from this repository's verify/gf2_fast extension. Divisor assembly, coset selection, the information-set draws, the orbit fold and the witness mirror are session tooling outside this repository, described above in enough detail to rewrite. Roughly two core-hours for the gate, five for the 8,000,000-trial run on a slower host, and a few minutes each for the screen and the fold.

Reproduction

m = 171, n = 2m = 342. With a = {67, 71, 118, 165, 169} and b = {13, 14, 15, 49, 58, 79, 97, 132, 159} as exponent sets,

H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]

where circ(s)[i, (i + e) mod m] = 1 for each e in s. The divisor is recovered as gcd(a, b, x^171 - 1), of degree 20. The submitted codes/342-40-32.json carries both matrices row by row and both distance witnesses.

Parity checks

X-checks 171 (max weight 14) · Z-checks 171 (max weight 14)
H_X (171 checks, sparse supports)
[67, 71, 118, 165, 169, 184, 185, 186, 220, 229, 250, 268, 303, 330] [68, 72, 119, 166, 170, 185, 186, 187, 221, 230, 251, 269, 304, 331] [0, 69, 73, 120, 167, 186, 187, 188, 222, 231, 252, 270, 305, 332] [1, 70, 74, 121, 168, 187, 188, 189, 223, 232, 253, 271, 306, 333] [2, 71, 75, 122, 169, 188, 189, 190, 224, 233, 254, 272, 307, 334] [3, 72, 76, 123, 170, 189, 190, 191, 225, 234, 255, 273, 308, 335] [0, 4, 73, 77, 124, 190, 191, 192, 226, 235, 256, 274, 309, 336] [1, 5, 74, 78, 125, 191, 192, 193, 227, 236, 257, 275, 310, 337] [2, 6, 75, 79, 126, 192, 193, 194, 228, 237, 258, 276, 311, 338] [3, 7, 76, 80, 127, 193, 194, 195, 229, 238, 259, 277, 312, 339] [4, 8, 77, 81, 128, 194, 195, 196, 230, 239, 260, 278, 313, 340] [5, 9, 78, 82, 129, 195, 196, 197, 231, 240, 261, 279, 314, 341] [6, 10, 79, 83, 130, 171, 196, 197, 198, 232, 241, 262, 280, 315] [7, 11, 80, 84, 131, 172, 197, 198, 199, 233, 242, 263, 281, 316] [8, 12, 81, 85, 132, 173, 198, 199, 200, 234, 243, 264, 282, 317] [9, 13, 82, 86, 133, 174, 199, 200, 201, 235, 244, 265, 283, 318] [10, 14, 83, 87, 134, 175, 200, 201, 202, 236, 245, 266, 284, 319] [11, 15, 84, 88, 135, 176, 201, 202, 203, 237, 246, 267, 285, 320] [12, 16, 85, 89, 136, 177, 202, 203, 204, 238, 247, 268, 286, 321] [13, 17, 86, 90, 137, 178, 203, 204, 205, 239, 248, 269, 287, 322] [14, 18, 87, 91, 138, 179, 204, 205, 206, 240, 249, 270, 288, 323] [15, 19, 88, 92, 139, 180, 205, 206, 207, 241, 250, 271, 289, 324] [16, 20, 89, 93, 140, 181, 206, 207, 208, 242, 251, 272, 290, 325] [17, 21, 90, 94, 141, 182, 207, 208, 209, 243, 252, 273, 291, 326] [18, 22, 91, 95, 142, 183, 208, 209, 210, 244, 253, 274, 292, 327] [19, 23, 92, 96, 143, 184, 209, 210, 211, 245, 254, 275, 293, 328] [20, 24, 93, 97, 144, 185, 210, 211, 212, 246, 255, 276, 294, 329] [21, 25, 94, 98, 145, 186, 211, 212, 213, 247, 256, 277, 295, 330] [22, 26, 95, 99, 146, 187, 212, 213, 214, 248, 257, 278, 296, 331] [23, 27, 96, 100, 147, 188, 213, 214, 215, 249, 258, 279, 297, 332] [24, 28, 97, 101, 148, 189, 214, 215, 216, 250, 259, 280, 298, 333] [25, 29, 98, 102, 149, 190, 215, 216, 217, 251, 260, 281, 299, 334] [26, 30, 99, 103, 150, 191, 216, 217, 218, 252, 261, 282, 300, 335] [27, 31, 100, 104, 151, 192, 217, 218, 219, 253, 262, 283, 301, 336] [28, 32, 101, 105, 152, 193, 218, 219, 220, 254, 263, 284, 302, 337] [29, 33, 102, 106, 153, 194, 219, 220, 221, 255, 264, 285, 303, 338] [30, 34, 103, 107, 154, 195, 220, 221, 222, 256, 265, 286, 304, 339] [31, 35, 104, 108, 155, 196, 221, 222, 223, 257, 266, 287, 305, 340] [32, 36, 105, 109, 156, 197, 222, 223, 224, 258, 267, 288, 306, 341] [33, 37, 106, 110, 157, 171, 198, 223, 224, 225, 259, 268, 289, 307] [34, 38, 107, 111, 158, 172, 199, 224, 225, 226, 260, 269, 290, 308] [35, 39, 108, 112, 159, 173, 200, 225, 226, 227, 261, 270, 291, 309] [36, 40, 109, 113, 160, 174, 201, 226, 227, 228, 262, 271, 292, 310] [37, 41, 110, 114, 161, 175, 202, 227, 228, 229, 263, 272, 293, 311] [38, 42, 111, 115, 162, 176, 203, 228, 229, 230, 264, 273, 294, 312] [39, 43, 112, 116, 163, 177, 204, 229, 230, 231, 265, 274, 295, 313] [40, 44, 113, 117, 164, 178, 205, 230, 231, 232, 266, 275, 296, 314] [41, 45, 114, 118, 165, 179, 206, 231, 232, 233, 267, 276, 297, 315] [42, 46, 115, 119, 166, 180, 207, 232, 233, 234, 268, 277, 298, 316] [43, 47, 116, 120, 167, 181, 208, 233, 234, 235, 269, 278, 299, 317] [44, 48, 117, 121, 168, 182, 209, 234, 235, 236, 270, 279, 300, 318] [45, 49, 118, 122, 169, 183, 210, 235, 236, 237, 271, 280, 301, 319] [46, 50, 119, 123, 170, 184, 211, 236, 237, 238, 272, 281, 302, 320] [0, 47, 51, 120, 124, 185, 212, 237, 238, 239, 273, 282, 303, 321] [1, 48, 52, 121, 125, 186, 213, 238, 239, 240, 274, 283, 304, 322] [2, 49, 53, 122, 126, 187, 214, 239, 240, 241, 275, 284, 305, 323] [3, 50, 54, 123, 127, 188, 215, 240, 241, 242, 276, 285, 306, 324] [4, 51, 55, 124, 128, 189, 216, 241, 242, 243, 277, 286, 307, 325] [5, 52, 56, 125, 129, 190, 217, 242, 243, 244, 278, 287, 308, 326] [6, 53, 57, 126, 130, 191, 218, 243, 244, 245, 279, 288, 309, 327] [7, 54, 58, 127, 131, 192, 219, 244, 245, 246, 280, 289, 310, 328] [8, 55, 59, 128, 132, 193, 220, 245, 246, 247, 281, 290, 311, 329] [9, 56, 60, 129, 133, 194, 221, 246, 247, 248, 282, 291, 312, 330] [10, 57, 61, 130, 134, 195, 222, 247, 248, 249, 283, 292, 313, 331] [11, 58, 62, 131, 135, 196, 223, 248, 249, 250, 284, 293, 314, 332] [12, 59, 63, 132, 136, 197, 224, 249, 250, 251, 285, 294, 315, 333] [13, 60, 64, 133, 137, 198, 225, 250, 251, 252, 286, 295, 316, 334] [14, 61, 65, 134, 138, 199, 226, 251, 252, 253, 287, 296, 317, 335] [15, 62, 66, 135, 139, 200, 227, 252, 253, 254, 288, 297, 318, 336] [16, 63, 67, 136, 140, 201, 228, 253, 254, 255, 289, 298, 319, 337] [17, 64, 68, 137, 141, 202, 229, 254, 255, 256, 290, 299, 320, 338] [18, 65, 69, 138, 142, 203, 230, 255, 256, 257, 291, 300, 321, 339] [19, 66, 70, 139, 143, 204, 231, 256, 257, 258, 292, 301, 322, 340] [20, 67, 71, 140, 144, 205, 232, 257, 258, 259, 293, 302, 323, 341] [21, 68, 72, 141, 145, 171, 206, 233, 258, 259, 260, 294, 303, 324] [22, 69, 73, 142, 146, 172, 207, 234, 259, 260, 261, 295, 304, 325] [23, 70, 74, 143, 147, 173, 208, 235, 260, 261, 262, 296, 305, 326] [24, 71, 75, 144, 148, 174, 209, 236, 261, 262, 263, 297, 306, 327] [25, 72, 76, 145, 149, 175, 210, 237, 262, 263, 264, 298, 307, 328] [26, 73, 77, 146, 150, 176, 211, 238, 263, 264, 265, 299, 308, 329] [27, 74, 78, 147, 151, 177, 212, 239, 264, 265, 266, 300, 309, 330] [28, 75, 79, 148, 152, 178, 213, 240, 265, 266, 267, 301, 310, 331] [29, 76, 80, 149, 153, 179, 214, 241, 266, 267, 268, 302, 311, 332] [30, 77, 81, 150, 154, 180, 215, 242, 267, 268, 269, 303, 312, 333] [31, 78, 82, 151, 155, 181, 216, 243, 268, 269, 270, 304, 313, 334] [32, 79, 83, 152, 156, 182, 217, 244, 269, 270, 271, 305, 314, 335] [33, 80, 84, 153, 157, 183, 218, 245, 270, 271, 272, 306, 315, 336] [34, 81, 85, 154, 158, 184, 219, 246, 271, 272, 273, 307, 316, 337] [35, 82, 86, 155, 159, 185, 220, 247, 272, 273, 274, 308, 317, 338] [36, 83, 87, 156, 160, 186, 221, 248, 273, 274, 275, 309, 318, 339] [37, 84, 88, 157, 161, 187, 222, 249, 274, 275, 276, 310, 319, 340] [38, 85, 89, 158, 162, 188, 223, 250, 275, 276, 277, 311, 320, 341] [39, 86, 90, 159, 163, 171, 189, 224, 251, 276, 277, 278, 312, 321] [40, 87, 91, 160, 164, 172, 190, 225, 252, 277, 278, 279, 313, 322] [41, 88, 92, 161, 165, 173, 191, 226, 253, 278, 279, 280, 314, 323] [42, 89, 93, 162, 166, 174, 192, 227, 254, 279, 280, 281, 315, 324] [43, 90, 94, 163, 167, 175, 193, 228, 255, 280, 281, 282, 316, 325] [44, 91, 95, 164, 168, 176, 194, 229, 256, 281, 282, 283, 317, 326] [45, 92, 96, 165, 169, 177, 195, 230, 257, 282, 283, 284, 318, 327] [46, 93, 97, 166, 170, 178, 196, 231, 258, 283, 284, 285, 319, 328] [0, 47, 94, 98, 167, 179, 197, 232, 259, 284, 285, 286, 320, 329] [1, 48, 95, 99, 168, 180, 198, 233, 260, 285, 286, 287, 321, 330] [2, 49, 96, 100, 169, 181, 199, 234, 261, 286, 287, 288, 322, 331] [3, 50, 97, 101, 170, 182, 200, 235, 262, 287, 288, 289, 323, 332] [0, 4, 51, 98, 102, 183, 201, 236, 263, 288, 289, 290, 324, 333] [1, 5, 52, 99, 103, 184, 202, 237, 264, 289, 290, 291, 325, 334] [2, 6, 53, 100, 104, 185, 203, 238, 265, 290, 291, 292, 326, 335] [3, 7, 54, 101, 105, 186, 204, 239, 266, 291, 292, 293, 327, 336] [4, 8, 55, 102, 106, 187, 205, 240, 267, 292, 293, 294, 328, 337] [5, 9, 56, 103, 107, 188, 206, 241, 268, 293, 294, 295, 329, 338] [6, 10, 57, 104, 108, 189, 207, 242, 269, 294, 295, 296, 330, 339] [7, 11, 58, 105, 109, 190, 208, 243, 270, 295, 296, 297, 331, 340] [8, 12, 59, 106, 110, 191, 209, 244, 271, 296, 297, 298, 332, 341] [9, 13, 60, 107, 111, 171, 192, 210, 245, 272, 297, 298, 299, 333] [10, 14, 61, 108, 112, 172, 193, 211, 246, 273, 298, 299, 300, 334] [11, 15, 62, 109, 113, 173, 194, 212, 247, 274, 299, 300, 301, 335] [12, 16, 63, 110, 114, 174, 195, 213, 248, 275, 300, 301, 302, 336] [13, 17, 64, 111, 115, 175, 196, 214, 249, 276, 301, 302, 303, 337] [14, 18, 65, 112, 116, 176, 197, 215, 250, 277, 302, 303, 304, 338] [15, 19, 66, 113, 117, 177, 198, 216, 251, 278, 303, 304, 305, 339] [16, 20, 67, 114, 118, 178, 199, 217, 252, 279, 304, 305, 306, 340] [17, 21, 68, 115, 119, 179, 200, 218, 253, 280, 305, 306, 307, 341] [18, 22, 69, 116, 120, 171, 180, 201, 219, 254, 281, 306, 307, 308] [19, 23, 70, 117, 121, 172, 181, 202, 220, 255, 282, 307, 308, 309] [20, 24, 71, 118, 122, 173, 182, 203, 221, 256, 283, 308, 309, 310] [21, 25, 72, 119, 123, 174, 183, 204, 222, 257, 284, 309, 310, 311] [22, 26, 73, 120, 124, 175, 184, 205, 223, 258, 285, 310, 311, 312] [23, 27, 74, 121, 125, 176, 185, 206, 224, 259, 286, 311, 312, 313] [24, 28, 75, 122, 126, 177, 186, 207, 225, 260, 287, 312, 313, 314] [25, 29, 76, 123, 127, 178, 187, 208, 226, 261, 288, 313, 314, 315] [26, 30, 77, 124, 128, 179, 188, 209, 227, 262, 289, 314, 315, 316] [27, 31, 78, 125, 129, 180, 189, 210, 228, 263, 290, 315, 316, 317] [28, 32, 79, 126, 130, 181, 190, 211, 229, 264, 291, 316, 317, 318] [29, 33, 80, 127, 131, 182, 191, 212, 230, 265, 292, 317, 318, 319] [30, 34, 81, 128, 132, 183, 192, 213, 231, 266, 293, 318, 319, 320] [31, 35, 82, 129, 133, 184, 193, 214, 232, 267, 294, 319, 320, 321] [32, 36, 83, 130, 134, 185, 194, 215, 233, 268, 295, 320, 321, 322] [33, 37, 84, 131, 135, 186, 195, 216, 234, 269, 296, 321, 322, 323] [34, 38, 85, 132, 136, 187, 196, 217, 235, 270, 297, 322, 323, 324] [35, 39, 86, 133, 137, 188, 197, 218, 236, 271, 298, 323, 324, 325] [36, 40, 87, 134, 138, 189, 198, 219, 237, 272, 299, 324, 325, 326] [37, 41, 88, 135, 139, 190, 199, 220, 238, 273, 300, 325, 326, 327] [38, 42, 89, 136, 140, 191, 200, 221, 239, 274, 301, 326, 327, 328] [39, 43, 90, 137, 141, 192, 201, 222, 240, 275, 302, 327, 328, 329] [40, 44, 91, 138, 142, 193, 202, 223, 241, 276, 303, 328, 329, 330] [41, 45, 92, 139, 143, 194, 203, 224, 242, 277, 304, 329, 330, 331] [42, 46, 93, 140, 144, 195, 204, 225, 243, 278, 305, 330, 331, 332] [43, 47, 94, 141, 145, 196, 205, 226, 244, 279, 306, 331, 332, 333] [44, 48, 95, 142, 146, 197, 206, 227, 245, 280, 307, 332, 333, 334] [45, 49, 96, 143, 147, 198, 207, 228, 246, 281, 308, 333, 334, 335] [46, 50, 97, 144, 148, 199, 208, 229, 247, 282, 309, 334, 335, 336] [47, 51, 98, 145, 149, 200, 209, 230, 248, 283, 310, 335, 336, 337] [48, 52, 99, 146, 150, 201, 210, 231, 249, 284, 311, 336, 337, 338] [49, 53, 100, 147, 151, 202, 211, 232, 250, 285, 312, 337, 338, 339] [50, 54, 101, 148, 152, 203, 212, 233, 251, 286, 313, 338, 339, 340] [51, 55, 102, 149, 153, 204, 213, 234, 252, 287, 314, 339, 340, 341] [52, 56, 103, 150, 154, 171, 205, 214, 235, 253, 288, 315, 340, 341] [53, 57, 104, 151, 155, 171, 172, 206, 215, 236, 254, 289, 316, 341] [54, 58, 105, 152, 156, 171, 172, 173, 207, 216, 237, 255, 290, 317] [55, 59, 106, 153, 157, 172, 173, 174, 208, 217, 238, 256, 291, 318] [56, 60, 107, 154, 158, 173, 174, 175, 209, 218, 239, 257, 292, 319] [57, 61, 108, 155, 159, 174, 175, 176, 210, 219, 240, 258, 293, 320] [58, 62, 109, 156, 160, 175, 176, 177, 211, 220, 241, 259, 294, 321] [59, 63, 110, 157, 161, 176, 177, 178, 212, 221, 242, 260, 295, 322] [60, 64, 111, 158, 162, 177, 178, 179, 213, 222, 243, 261, 296, 323] [61, 65, 112, 159, 163, 178, 179, 180, 214, 223, 244, 262, 297, 324] [62, 66, 113, 160, 164, 179, 180, 181, 215, 224, 245, 263, 298, 325] [63, 67, 114, 161, 165, 180, 181, 182, 216, 225, 246, 264, 299, 326] [64, 68, 115, 162, 166, 181, 182, 183, 217, 226, 247, 265, 300, 327] [65, 69, 116, 163, 167, 182, 183, 184, 218, 227, 248, 266, 301, 328] [66, 70, 117, 164, 168, 183, 184, 185, 219, 228, 249, 267, 302, 329]
H_Z (171 checks, sparse supports)
[12, 39, 74, 92, 113, 122, 156, 157, 158, 173, 177, 224, 271, 275] [13, 40, 75, 93, 114, 123, 157, 158, 159, 174, 178, 225, 272, 276] [14, 41, 76, 94, 115, 124, 158, 159, 160, 175, 179, 226, 273, 277] [15, 42, 77, 95, 116, 125, 159, 160, 161, 176, 180, 227, 274, 278] [16, 43, 78, 96, 117, 126, 160, 161, 162, 177, 181, 228, 275, 279] [17, 44, 79, 97, 118, 127, 161, 162, 163, 178, 182, 229, 276, 280] [18, 45, 80, 98, 119, 128, 162, 163, 164, 179, 183, 230, 277, 281] [19, 46, 81, 99, 120, 129, 163, 164, 165, 180, 184, 231, 278, 282] [20, 47, 82, 100, 121, 130, 164, 165, 166, 181, 185, 232, 279, 283] [21, 48, 83, 101, 122, 131, 165, 166, 167, 182, 186, 233, 280, 284] [22, 49, 84, 102, 123, 132, 166, 167, 168, 183, 187, 234, 281, 285] [23, 50, 85, 103, 124, 133, 167, 168, 169, 184, 188, 235, 282, 286] [24, 51, 86, 104, 125, 134, 168, 169, 170, 185, 189, 236, 283, 287] [0, 25, 52, 87, 105, 126, 135, 169, 170, 186, 190, 237, 284, 288] [0, 1, 26, 53, 88, 106, 127, 136, 170, 187, 191, 238, 285, 289] [0, 1, 2, 27, 54, 89, 107, 128, 137, 188, 192, 239, 286, 290] [1, 2, 3, 28, 55, 90, 108, 129, 138, 189, 193, 240, 287, 291] [2, 3, 4, 29, 56, 91, 109, 130, 139, 190, 194, 241, 288, 292] [3, 4, 5, 30, 57, 92, 110, 131, 140, 191, 195, 242, 289, 293] [4, 5, 6, 31, 58, 93, 111, 132, 141, 192, 196, 243, 290, 294] [5, 6, 7, 32, 59, 94, 112, 133, 142, 193, 197, 244, 291, 295] [6, 7, 8, 33, 60, 95, 113, 134, 143, 194, 198, 245, 292, 296] [7, 8, 9, 34, 61, 96, 114, 135, 144, 195, 199, 246, 293, 297] [8, 9, 10, 35, 62, 97, 115, 136, 145, 196, 200, 247, 294, 298] [9, 10, 11, 36, 63, 98, 116, 137, 146, 197, 201, 248, 295, 299] [10, 11, 12, 37, 64, 99, 117, 138, 147, 198, 202, 249, 296, 300] [11, 12, 13, 38, 65, 100, 118, 139, 148, 199, 203, 250, 297, 301] [12, 13, 14, 39, 66, 101, 119, 140, 149, 200, 204, 251, 298, 302] [13, 14, 15, 40, 67, 102, 120, 141, 150, 201, 205, 252, 299, 303] [14, 15, 16, 41, 68, 103, 121, 142, 151, 202, 206, 253, 300, 304] [15, 16, 17, 42, 69, 104, 122, 143, 152, 203, 207, 254, 301, 305] [16, 17, 18, 43, 70, 105, 123, 144, 153, 204, 208, 255, 302, 306] [17, 18, 19, 44, 71, 106, 124, 145, 154, 205, 209, 256, 303, 307] [18, 19, 20, 45, 72, 107, 125, 146, 155, 206, 210, 257, 304, 308] [19, 20, 21, 46, 73, 108, 126, 147, 156, 207, 211, 258, 305, 309] [20, 21, 22, 47, 74, 109, 127, 148, 157, 208, 212, 259, 306, 310] [21, 22, 23, 48, 75, 110, 128, 149, 158, 209, 213, 260, 307, 311] [22, 23, 24, 49, 76, 111, 129, 150, 159, 210, 214, 261, 308, 312] [23, 24, 25, 50, 77, 112, 130, 151, 160, 211, 215, 262, 309, 313] [24, 25, 26, 51, 78, 113, 131, 152, 161, 212, 216, 263, 310, 314] [25, 26, 27, 52, 79, 114, 132, 153, 162, 213, 217, 264, 311, 315] [26, 27, 28, 53, 80, 115, 133, 154, 163, 214, 218, 265, 312, 316] [27, 28, 29, 54, 81, 116, 134, 155, 164, 215, 219, 266, 313, 317] [28, 29, 30, 55, 82, 117, 135, 156, 165, 216, 220, 267, 314, 318] [29, 30, 31, 56, 83, 118, 136, 157, 166, 217, 221, 268, 315, 319] [30, 31, 32, 57, 84, 119, 137, 158, 167, 218, 222, 269, 316, 320] [31, 32, 33, 58, 85, 120, 138, 159, 168, 219, 223, 270, 317, 321] [32, 33, 34, 59, 86, 121, 139, 160, 169, 220, 224, 271, 318, 322] [33, 34, 35, 60, 87, 122, 140, 161, 170, 221, 225, 272, 319, 323] [0, 34, 35, 36, 61, 88, 123, 141, 162, 222, 226, 273, 320, 324] [1, 35, 36, 37, 62, 89, 124, 142, 163, 223, 227, 274, 321, 325] [2, 36, 37, 38, 63, 90, 125, 143, 164, 224, 228, 275, 322, 326] [3, 37, 38, 39, 64, 91, 126, 144, 165, 225, 229, 276, 323, 327] [4, 38, 39, 40, 65, 92, 127, 145, 166, 226, 230, 277, 324, 328] [5, 39, 40, 41, 66, 93, 128, 146, 167, 227, 231, 278, 325, 329] [6, 40, 41, 42, 67, 94, 129, 147, 168, 228, 232, 279, 326, 330] [7, 41, 42, 43, 68, 95, 130, 148, 169, 229, 233, 280, 327, 331] [8, 42, 43, 44, 69, 96, 131, 149, 170, 230, 234, 281, 328, 332] [0, 9, 43, 44, 45, 70, 97, 132, 150, 231, 235, 282, 329, 333] [1, 10, 44, 45, 46, 71, 98, 133, 151, 232, 236, 283, 330, 334] [2, 11, 45, 46, 47, 72, 99, 134, 152, 233, 237, 284, 331, 335] [3, 12, 46, 47, 48, 73, 100, 135, 153, 234, 238, 285, 332, 336] [4, 13, 47, 48, 49, 74, 101, 136, 154, 235, 239, 286, 333, 337] [5, 14, 48, 49, 50, 75, 102, 137, 155, 236, 240, 287, 334, 338] [6, 15, 49, 50, 51, 76, 103, 138, 156, 237, 241, 288, 335, 339] [7, 16, 50, 51, 52, 77, 104, 139, 157, 238, 242, 289, 336, 340] [8, 17, 51, 52, 53, 78, 105, 140, 158, 239, 243, 290, 337, 341] [9, 18, 52, 53, 54, 79, 106, 141, 159, 171, 240, 244, 291, 338] [10, 19, 53, 54, 55, 80, 107, 142, 160, 172, 241, 245, 292, 339] [11, 20, 54, 55, 56, 81, 108, 143, 161, 173, 242, 246, 293, 340] [12, 21, 55, 56, 57, 82, 109, 144, 162, 174, 243, 247, 294, 341] [13, 22, 56, 57, 58, 83, 110, 145, 163, 171, 175, 244, 248, 295] [14, 23, 57, 58, 59, 84, 111, 146, 164, 172, 176, 245, 249, 296] [15, 24, 58, 59, 60, 85, 112, 147, 165, 173, 177, 246, 250, 297] [16, 25, 59, 60, 61, 86, 113, 148, 166, 174, 178, 247, 251, 298] [17, 26, 60, 61, 62, 87, 114, 149, 167, 175, 179, 248, 252, 299] [18, 27, 61, 62, 63, 88, 115, 150, 168, 176, 180, 249, 253, 300] [19, 28, 62, 63, 64, 89, 116, 151, 169, 177, 181, 250, 254, 301] [20, 29, 63, 64, 65, 90, 117, 152, 170, 178, 182, 251, 255, 302] [0, 21, 30, 64, 65, 66, 91, 118, 153, 179, 183, 252, 256, 303] [1, 22, 31, 65, 66, 67, 92, 119, 154, 180, 184, 253, 257, 304] [2, 23, 32, 66, 67, 68, 93, 120, 155, 181, 185, 254, 258, 305] [3, 24, 33, 67, 68, 69, 94, 121, 156, 182, 186, 255, 259, 306] [4, 25, 34, 68, 69, 70, 95, 122, 157, 183, 187, 256, 260, 307] [5, 26, 35, 69, 70, 71, 96, 123, 158, 184, 188, 257, 261, 308] [6, 27, 36, 70, 71, 72, 97, 124, 159, 185, 189, 258, 262, 309] [7, 28, 37, 71, 72, 73, 98, 125, 160, 186, 190, 259, 263, 310] [8, 29, 38, 72, 73, 74, 99, 126, 161, 187, 191, 260, 264, 311] [9, 30, 39, 73, 74, 75, 100, 127, 162, 188, 192, 261, 265, 312] [10, 31, 40, 74, 75, 76, 101, 128, 163, 189, 193, 262, 266, 313] [11, 32, 41, 75, 76, 77, 102, 129, 164, 190, 194, 263, 267, 314] [12, 33, 42, 76, 77, 78, 103, 130, 165, 191, 195, 264, 268, 315] [13, 34, 43, 77, 78, 79, 104, 131, 166, 192, 196, 265, 269, 316] [14, 35, 44, 78, 79, 80, 105, 132, 167, 193, 197, 266, 270, 317] [15, 36, 45, 79, 80, 81, 106, 133, 168, 194, 198, 267, 271, 318] [16, 37, 46, 80, 81, 82, 107, 134, 169, 195, 199, 268, 272, 319] [17, 38, 47, 81, 82, 83, 108, 135, 170, 196, 200, 269, 273, 320] [0, 18, 39, 48, 82, 83, 84, 109, 136, 197, 201, 270, 274, 321] [1, 19, 40, 49, 83, 84, 85, 110, 137, 198, 202, 271, 275, 322] [2, 20, 41, 50, 84, 85, 86, 111, 138, 199, 203, 272, 276, 323] [3, 21, 42, 51, 85, 86, 87, 112, 139, 200, 204, 273, 277, 324] [4, 22, 43, 52, 86, 87, 88, 113, 140, 201, 205, 274, 278, 325] [5, 23, 44, 53, 87, 88, 89, 114, 141, 202, 206, 275, 279, 326] [6, 24, 45, 54, 88, 89, 90, 115, 142, 203, 207, 276, 280, 327] [7, 25, 46, 55, 89, 90, 91, 116, 143, 204, 208, 277, 281, 328] [8, 26, 47, 56, 90, 91, 92, 117, 144, 205, 209, 278, 282, 329] [9, 27, 48, 57, 91, 92, 93, 118, 145, 206, 210, 279, 283, 330] [10, 28, 49, 58, 92, 93, 94, 119, 146, 207, 211, 280, 284, 331] [11, 29, 50, 59, 93, 94, 95, 120, 147, 208, 212, 281, 285, 332] [12, 30, 51, 60, 94, 95, 96, 121, 148, 209, 213, 282, 286, 333] [13, 31, 52, 61, 95, 96, 97, 122, 149, 210, 214, 283, 287, 334] [14, 32, 53, 62, 96, 97, 98, 123, 150, 211, 215, 284, 288, 335] [15, 33, 54, 63, 97, 98, 99, 124, 151, 212, 216, 285, 289, 336] [16, 34, 55, 64, 98, 99, 100, 125, 152, 213, 217, 286, 290, 337] [17, 35, 56, 65, 99, 100, 101, 126, 153, 214, 218, 287, 291, 338] [18, 36, 57, 66, 100, 101, 102, 127, 154, 215, 219, 288, 292, 339] [19, 37, 58, 67, 101, 102, 103, 128, 155, 216, 220, 289, 293, 340] [20, 38, 59, 68, 102, 103, 104, 129, 156, 217, 221, 290, 294, 341] [21, 39, 60, 69, 103, 104, 105, 130, 157, 171, 218, 222, 291, 295] [22, 40, 61, 70, 104, 105, 106, 131, 158, 172, 219, 223, 292, 296] [23, 41, 62, 71, 105, 106, 107, 132, 159, 173, 220, 224, 293, 297] [24, 42, 63, 72, 106, 107, 108, 133, 160, 174, 221, 225, 294, 298] [25, 43, 64, 73, 107, 108, 109, 134, 161, 175, 222, 226, 295, 299] [26, 44, 65, 74, 108, 109, 110, 135, 162, 176, 223, 227, 296, 300] [27, 45, 66, 75, 109, 110, 111, 136, 163, 177, 224, 228, 297, 301] [28, 46, 67, 76, 110, 111, 112, 137, 164, 178, 225, 229, 298, 302] [29, 47, 68, 77, 111, 112, 113, 138, 165, 179, 226, 230, 299, 303] [30, 48, 69, 78, 112, 113, 114, 139, 166, 180, 227, 231, 300, 304] [31, 49, 70, 79, 113, 114, 115, 140, 167, 181, 228, 232, 301, 305] [32, 50, 71, 80, 114, 115, 116, 141, 168, 182, 229, 233, 302, 306] [33, 51, 72, 81, 115, 116, 117, 142, 169, 183, 230, 234, 303, 307] [34, 52, 73, 82, 116, 117, 118, 143, 170, 184, 231, 235, 304, 308] [0, 35, 53, 74, 83, 117, 118, 119, 144, 185, 232, 236, 305, 309] [1, 36, 54, 75, 84, 118, 119, 120, 145, 186, 233, 237, 306, 310] [2, 37, 55, 76, 85, 119, 120, 121, 146, 187, 234, 238, 307, 311] [3, 38, 56, 77, 86, 120, 121, 122, 147, 188, 235, 239, 308, 312] [4, 39, 57, 78, 87, 121, 122, 123, 148, 189, 236, 240, 309, 313] [5, 40, 58, 79, 88, 122, 123, 124, 149, 190, 237, 241, 310, 314] [6, 41, 59, 80, 89, 123, 124, 125, 150, 191, 238, 242, 311, 315] [7, 42, 60, 81, 90, 124, 125, 126, 151, 192, 239, 243, 312, 316] [8, 43, 61, 82, 91, 125, 126, 127, 152, 193, 240, 244, 313, 317] [9, 44, 62, 83, 92, 126, 127, 128, 153, 194, 241, 245, 314, 318] [10, 45, 63, 84, 93, 127, 128, 129, 154, 195, 242, 246, 315, 319] [11, 46, 64, 85, 94, 128, 129, 130, 155, 196, 243, 247, 316, 320] [12, 47, 65, 86, 95, 129, 130, 131, 156, 197, 244, 248, 317, 321] [13, 48, 66, 87, 96, 130, 131, 132, 157, 198, 245, 249, 318, 322] [14, 49, 67, 88, 97, 131, 132, 133, 158, 199, 246, 250, 319, 323] [15, 50, 68, 89, 98, 132, 133, 134, 159, 200, 247, 251, 320, 324] [16, 51, 69, 90, 99, 133, 134, 135, 160, 201, 248, 252, 321, 325] [17, 52, 70, 91, 100, 134, 135, 136, 161, 202, 249, 253, 322, 326] [18, 53, 71, 92, 101, 135, 136, 137, 162, 203, 250, 254, 323, 327] [19, 54, 72, 93, 102, 136, 137, 138, 163, 204, 251, 255, 324, 328] [20, 55, 73, 94, 103, 137, 138, 139, 164, 205, 252, 256, 325, 329] [21, 56, 74, 95, 104, 138, 139, 140, 165, 206, 253, 257, 326, 330] [22, 57, 75, 96, 105, 139, 140, 141, 166, 207, 254, 258, 327, 331] [23, 58, 76, 97, 106, 140, 141, 142, 167, 208, 255, 259, 328, 332] [24, 59, 77, 98, 107, 141, 142, 143, 168, 209, 256, 260, 329, 333] [25, 60, 78, 99, 108, 142, 143, 144, 169, 210, 257, 261, 330, 334] [26, 61, 79, 100, 109, 143, 144, 145, 170, 211, 258, 262, 331, 335] [0, 27, 62, 80, 101, 110, 144, 145, 146, 212, 259, 263, 332, 336] [1, 28, 63, 81, 102, 111, 145, 146, 147, 213, 260, 264, 333, 337] [2, 29, 64, 82, 103, 112, 146, 147, 148, 214, 261, 265, 334, 338] [3, 30, 65, 83, 104, 113, 147, 148, 149, 215, 262, 266, 335, 339] [4, 31, 66, 84, 105, 114, 148, 149, 150, 216, 263, 267, 336, 340] [5, 32, 67, 85, 106, 115, 149, 150, 151, 217, 264, 268, 337, 341] [6, 33, 68, 86, 107, 116, 150, 151, 152, 171, 218, 265, 269, 338] [7, 34, 69, 87, 108, 117, 151, 152, 153, 172, 219, 266, 270, 339] [8, 35, 70, 88, 109, 118, 152, 153, 154, 173, 220, 267, 271, 340] [9, 36, 71, 89, 110, 119, 153, 154, 155, 174, 221, 268, 272, 341] [10, 37, 72, 90, 111, 120, 154, 155, 156, 171, 175, 222, 269, 273] [11, 38, 73, 91, 112, 121, 155, 156, 157, 172, 176, 223, 270, 274]
Code ID 342-40-32 · download JSON · raw on GitHub