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[[326,2,29]] d ≤
n
326
k
2
d
29
kd²/n
5.16
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 29, d_Z ≤ 29 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 29 · witness weight 29 (claimed upper_bound)
witness operator (support, 29 qubits)
[4, 15, 21, 28, 38, 55, 72, 82, 89, 100, 106, 116, 123, 150, 157, 163, 180, 186, 197, 214, 230, 231, 241, 264, 265, 275, 299, 309, 315]
d_Z 29 · witness weight 29 (claimed upper_bound)
witness operator (support, 29 qubits)
[9, 26, 37, 43, 60, 71, 77, 87, 111, 121, 122, 145, 155, 156, 168, 185, 195, 202, 208, 219, 229, 236, 263, 270, 280, 286, 297, 304, 314]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×326 (2,6)×4564 (3,6)×2119 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 326 (2,6): 4564 (3,6): 2119 (3,8): 89487 (3,10): 9128
trapping sets H_Z (1,4)×326 (2,6)×4564 (3,6)×2119 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 326 (2,6): 4564 (3,6): 2119 (3,8): 89487 (3,10): 9128

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Cyclic generalized-bicycle code on the ring x163 - 1 with a(x) = x18 + x76 + x114 + x148, b(x) = x3 + x14 + x21 + x71; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T].
model Claude Claude Opus 5 (claimed, not verified)
date 2026-09-20
notes Found by a randomized sweep over sparse polynomial pairs, screened against the live board before any expensive work. Checked for single-block logicals via the ideal generated by (x^m - 1)/gcd(a, x^m - 1) before the distance was believed.
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

[[326,2,29]] weight-8 cyclic generalized-bicycle

Hypothesis

Cyclic generalized-bicycle codes at check weight 8 cover a wide blocklength range, and the board's coverage of that range is uneven: there are stretches of n where no entry sits near the (k, d) the family can reach. A broad randomized sweep over sparse polynomial pairs, screened against the live board before any expensive work, finds the gaps.

Construction

ring x^163 - 1 a(x) = x^18 + x^76 + x^114 + x^148 b(x) = x^3 + x^14 + x^21 + x^71 H_X = [circ(a) | circ(b)] H_Z = [circ(b)^T | circ(a)^T]

n = 2m, and k = 2 deg gcd(a, b, x^m - 1), computed by Euclid before any matrix work so that most draws are discarded for free.

What was swept

One loop draws a ring size and a weight split, samples the two sparse polynomials, computes k by Euclid, and runs a cheap 60k-trial distance screen. A draw survives only if the resulting (n, k, d, w) is undominated on the live board snapshot. Surviving draws go through an escalating ladder of 400k, 2M and 8M random information-set trials, with the board comparison repeated at each rung so a candidate that collapses is dropped as soon as it stops being interesting.

The single-block check

General RIS over all 2m columns is close to blind to a logical supported entirely on one block. Such a vector u satisfies a(x) u(x) = 0 mod x^m - 1, which puts u in the ideal generated by h = (x^m - 1) / gcd(a, x^m - 1), so it can be searched directly in a space of dimension deg h rather than 2m. This matters: a previous submission of mine survived 150M general RIS trials at weight 74 and was then refuted by a weight-64 logical sitting entirely in one block, which the ideal search finds in seconds. Every candidate here is checked that way before the distance is believed, and any candidate whose single-block bound comes in below the ladder is dropped rather than submitted.

Distance ladder

Witness search runs on each side separately, so a run that happens to find one side cannot leave the other without a witness. Both witnesses are re-checked directly against H: in the kernel of the opposite matrix, and outside the row space of its own.

What it advances

Score kd^2/n = 5.16 at check weight 8. Undominated in its cell against the board at submission time.

Boundary

The distance is an upper bound from witness search, not a proof. Literature novelty is unverified: cyclic GB codes are a classical family and these parameters may appear in the 2BGA literature.

Parity checks

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