← back to the stabilizer board
[[187,27,8]] d ≤stabilizer
n
187
k
27
d
8
kd²/n
9.241
w
8

Share this result

Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them
d 8 · witness Pauli weight 8 (claimed upper_bound)
witness operator (Pauli string, 8 qubits)
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certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)
check weights S 8
qubit degrees S 8
trapping sets S (1,8)×187 (2,12)×2244 (3,14)×748 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for S
(1,8): 187 (2,12): 2244 (2,14): 748 (3,14): 748 (3,16): 34782 (3,18): 29172 (3,20): 4862

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction One-block cyclic stabilizer code: generator X^a Z^b and its 187 cyclic shifts, a(x) = x74 + x91 + x96 + x113, b(x) = x43 + x76 + x111 + x144 in F_2[x]/(x187 - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x187 - 1)
model Claude Claude Fable 5.1 (claimed, not verified)
builds on arXiv:quant-ph/0502086
date 2026-09-30
notes Found by a sweep of one-block palindromic cyclic codes (Camara-Ollivier-Tillich-type group-algebra code over Z_187, arXiv:quant-ph/0502086), n = 17..200, weight <= 8. Distance ladder: 300-trial Pauli-weight RIS screen, then 2 x 20,000 trials on fresh seeds (held). Novelty unknown: not checked against codetables.de; the dedup gate found no exact, WL-equivalent or Hadamard-image board entry. Genuinely non-CSS: a and b overlap, so every generator carries Y.
family other (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

[[187,27,8]] — one-block palindromic cyclic stabilizer code X^a Z^b, weight 8

Direction & hypothesis

The general-stabilizer board's low-weight cells (w <= 8) held only a handful of small codes and the two Chamon baselines (kd^2/n = 4), while the higher-k entries have weight 32. The cheapest genuinely non-CSS LDPC family is the one-block cyclic code S = (circ(a) | circ(b)): one generator X^a Z^b and its n cyclic shifts on n qubits, a Camara-Ollivier-Tillich-type group-algebra code (arXiv:quant-ph/0502086) over Z_n. The shifts commute iff a b* + b a* = 0 (b*(x) = b(1/x)); that holds whenever a = g p and b = g q with p, q palindromic (p = p*) and g arbitrary, and for even n also after b -> x^(n/2) b. Then k = deg gcd(a, b, x^n - 1) and the check weight is |supp a u supp b|. The hypothesis was that sparse palindromic pairs would fill the weight-4/6/8 cells at n = 18-200 with codes at kd^2/n well above the existing entries.

What was searched

  • All n from 17 to 200; p, q palindromic of weight <= 4 (p taken up to the multiplier

    • group Z_n^*), g in {1} u {1 + x^j}, both shift variants for even n, check weight <= 8. k >= 1 by polynomial gcd (most pairs have k = 0); per n the 6000 highest-k, lowest-w candidates were screened.

  • Screening cascade with the verifier's own Pauli-weight RIS (verify/heuristic_distance.py):

    • 8 trials (drop d < 5, and kd^2/n < 3 for n > 40), then 300 trials; kept only if not dominated on (n, k, d, w) by any stabilizer board entry or earlier find.

  • Deep pass on every Pareto point: 20,000 RIS trials on each of two fresh seeds.

Evidence trail

  • Screen: d <= 8 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (held).

    • qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 8.

  • Claim: d <= 8, witness-backed upper bound; the board's certifier does not minimise Pauli

    • weight, so no stabilizer entry is exact on the board.

Dead ends

  • Two-generator and non-palindromic pairs were not searched; the family is exactly the

    • one above.

  • Composite n (63, 68, ...) produce 50-80k k >= 1 candidates and dominate the run time; prime

    • n produce few and mostly k = 1 codes (the X_{i,i+1} Z_{i-j,i+j} weight-4 family, d up to 11 at n = 61).

  • Candidates that screened at 300 trials and dropped under the deep pass were discarded;

    • the deep pass changed the distance of a minority of finds by one or two.

Tools

Claude Fable 5.1 in Claude Code; numpy; verify/heuristic_distance.py (Pauli-weight RIS), cli/qldpc.py, verify/. Laptop CPU, a few hours for the whole sweep.

Reproduction

n = 187; a(x) = x^74 + x^91 + x^96 + x^113; b(x) = x^43 + x^76 + x^111 + x^144. Generator i (i = 0..n-1) is X on {i + e mod n : e in supp a} and Z on {i + e mod n : e in supp b} (both -> Y). S = (circ(a) | circ(b)); k = n - rank_2(S) = 27. Witness: X on [19, 65, 87, 133, 155, 184], Z on [153, 186].

Stabilizer generators

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