[[171,25,7]] · QEC Challenge

[[171,25,7]] d ≤stabilizer

171

kd²/n

7.164

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 7 · witness Pauli weight 7 (claimed upper_bound)

witness operator (Pauli string, 7 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)×171 (2,8)×171 (3,8)×171 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for S

(1,8): 171 (2,8): 171 (2,10): 171 (2,12): 1539 (2,14): 513 (3,8): 171 (3,10): 342 (3,12): 4161 (3,14): 5472 (3,16): 17271 (3,18): 12996 (3,20): 1881

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 171 cyclic shifts, a(x) = 1 + x¹⁹ + x⁸⁴ + x⁸⁷ + x¹⁰³ + x¹⁰⁶, b(x) = x⁴⁹ + x⁸⁷ + x¹⁰³ + x¹⁴¹ in F_2[x]/(x¹⁷¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁷¹ - 1)

model 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_171, 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

[[171,25,7]] — 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 <= 7 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 7.

Claim: d <= 7, 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 = 171; a(x) = 1 + x^19 + x^84 + x^87 + x^103 + x^106; b(x) = x^49 + x^87 + x^103 + x^141. 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) = 25. Witness: X on [26, 110, 113], Z on [75, 94, 129, 148].

Stabilizer generators

generators 171 (max weight 8; 171 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 (171, Pauli strings on 171 qubits)

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IIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIX

symplectic rows (A | B) (171, sparse supports)

X: [0, 19, 84, 87, 103, 106] Z: [49, 87, 103, 141] X: [1, 20, 85, 88, 104, 107] Z: [50, 88, 104, 142] X: [2, 21, 86, 89, 105, 108] Z: [51, 89, 105, 143] X: [3, 22, 87, 90, 106, 109] Z: [52, 90, 106, 144] X: [4, 23, 88, 91, 107, 110] Z: [53, 91, 107, 145] X: [5, 24, 89, 92, 108, 111] Z: [54, 92, 108, 146] X: [6, 25, 90, 93, 109, 112] Z: [55, 93, 109, 147] X: [7, 26, 91, 94, 110, 113] Z: [56, 94, 110, 148] X: [8, 27, 92, 95, 111, 114] Z: [57, 95, 111, 149] X: [9, 28, 93, 96, 112, 115] Z: [58, 96, 112, 150] X: [10, 29, 94, 97, 113, 116] Z: [59, 97, 113, 151] X: [11, 30, 95, 98, 114, 117] Z: [60, 98, 114, 152] X: [12, 31, 96, 99, 115, 118] Z: [61, 99, 115, 153] X: [13, 32, 97, 100, 116, 119] Z: [62, 100, 116, 154] X: [14, 33, 98, 101, 117, 120] Z: [63, 101, 117, 155] X: [15, 34, 99, 102, 118, 121] Z: [64, 102, 118, 156] X: [16, 35, 100, 103, 119, 122] Z: [65, 103, 119, 157] X: [17, 36, 101, 104, 120, 123] Z: [66, 104, 120, 158] X: [18, 37, 102, 105, 121, 124] Z: [67, 105, 121, 159] X: [19, 38, 103, 106, 122, 125] Z: [68, 106, 122, 160] X: [20, 39, 104, 107, 123, 126] Z: [69, 107, 123, 161] X: [21, 40, 105, 108, 124, 127] Z: [70, 108, 124, 162] X: [22, 41, 106, 109, 125, 128] Z: [71, 109, 125, 163] X: [23, 42, 107, 110, 126, 129] Z: [72, 110, 126, 164] X: [24, 43, 108, 111, 127, 130] Z: [73, 111, 127, 165] X: [25, 44, 109, 112, 128, 131] Z: [74, 112, 128, 166] X: [26, 45, 110, 113, 129, 132] Z: [75, 113, 129, 167] X: [27, 46, 111, 114, 130, 133] Z: [76, 114, 130, 168] X: [28, 47, 112, 115, 131, 134] Z: [77, 115, 131, 169] X: [29, 48, 113, 116, 132, 135] Z: [78, 116, 132, 170] X: [30, 49, 114, 117, 133, 136] Z: [0, 79, 117, 133] X: [31, 50, 115, 118, 134, 137] Z: [1, 80, 118, 134] X: [32, 51, 116, 119, 135, 138] Z: [2, 81, 119, 135] X: [33, 52, 117, 120, 136, 139] Z: [3, 82, 120, 136] X: [34, 53, 118, 121, 137, 140] Z: [4, 83, 121, 137] X: [35, 54, 119, 122, 138, 141] Z: [5, 84, 122, 138] X: [36, 55, 120, 123, 139, 142] Z: [6, 85, 123, 139] X: [37, 56, 121, 124, 140, 143] Z: [7, 86, 124, 140] X: [38, 57, 122, 125, 141, 144] Z: [8, 87, 125, 141] X: [39, 58, 123, 126, 142, 145] Z: [9, 88, 126, 142] X: [40, 59, 124, 127, 143, 146] Z: [10, 89, 127, 143] X: [41, 60, 125, 128, 144, 147] Z: [11, 90, 128, 144] X: [42, 61, 126, 129, 145, 148] Z: [12, 91, 129, 145] X: [43, 62, 127, 130, 146, 149] Z: [13, 92, 130, 146] X: [44, 63, 128, 131, 147, 150] Z: [14, 93, 131, 147] X: [45, 64, 129, 132, 148, 151] Z: [15, 94, 132, 148] X: [46, 65, 130, 133, 149, 152] Z: [16, 95, 133, 149] X: [47, 66, 131, 134, 150, 153] Z: [17, 96, 134, 150] X: [48, 67, 132, 135, 151, 154] Z: [18, 97, 135, 151] X: [49, 68, 133, 136, 152, 155] Z: [19, 98, 136, 152] X: [50, 69, 134, 137, 153, 156] Z: [20, 99, 137, 153] X: [51, 70, 135, 138, 154, 157] Z: [21, 100, 138, 154] X: [52, 71, 136, 139, 155, 158] Z: [22, 101, 139, 155] X: [53, 72, 137, 140, 156, 159] Z: [23, 102, 140, 156] X: [54, 73, 138, 141, 157, 160] Z: [24, 103, 141, 157] X: [55, 74, 139, 142, 158, 161] Z: [25, 104, 142, 158] X: [56, 75, 140, 143, 159, 162] Z: [26, 105, 143, 159] X: [57, 76, 141, 144, 160, 163] Z: [27, 106, 144, 160] X: [58, 77, 142, 145, 161, 164] Z: [28, 107, 145, 161] X: [59, 78, 143, 146, 162, 165] Z: [29, 108, 146, 162] X: [60, 79, 144, 147, 163, 166] Z: [30, 109, 147, 163] X: [61, 80, 145, 148, 164, 167] Z: [31, 110, 148, 164] X: [62, 81, 146, 149, 165, 168] Z: [32, 111, 149, 165] X: [63, 82, 147, 150, 166, 169] Z: [33, 112, 150, 166] X: [64, 83, 148, 151, 167, 170] Z: [34, 113, 151, 167] X: [0, 65, 84, 149, 152, 168] Z: [35, 114, 152, 168] X: [1, 66, 85, 150, 153, 169] Z: [36, 115, 153, 169] X: [2, 67, 86, 151, 154, 170] Z: [37, 116, 154, 170] X: [0, 3, 68, 87, 152, 155] Z: [0, 38, 117, 155] X: [1, 4, 69, 88, 153, 156] Z: [1, 39, 118, 156] X: [2, 5, 70, 89, 154, 157] Z: [2, 40, 119, 157] X: [3, 6, 71, 90, 155, 158] Z: [3, 41, 120, 158] X: [4, 7, 72, 91, 156, 159] Z: [4, 42, 121, 159] X: [5, 8, 73, 92, 157, 160] Z: [5, 43, 122, 160] X: [6, 9, 74, 93, 158, 161] Z: [6, 44, 123, 161] X: [7, 10, 75, 94, 159, 162] Z: [7, 45, 124, 162] X: [8, 11, 76, 95, 160, 163] Z: [8, 46, 125, 163] X: [9, 12, 77, 96, 161, 164] Z: [9, 47, 126, 164] X: [10, 13, 78, 97, 162, 165] Z: [10, 48, 127, 165] X: [11, 14, 79, 98, 163, 166] Z: [11, 49, 128, 166] X: [12, 15, 80, 99, 164, 167] Z: [12, 50, 129, 167] X: [13, 16, 81, 100, 165, 168] Z: [13, 51, 130, 168] X: [14, 17, 82, 101, 166, 169] Z: [14, 52, 131, 169] X: [15, 18, 83, 102, 167, 170] Z: [15, 53, 132, 170] X: [0, 16, 19, 84, 103, 168] Z: [0, 16, 54, 133] X: [1, 17, 20, 85, 104, 169] Z: [1, 17, 55, 134] X: [2, 18, 21, 86, 105, 170] Z: [2, 18, 56, 135] X: [0, 3, 19, 22, 87, 106] Z: [3, 19, 57, 136] X: [1, 4, 20, 23, 88, 107] Z: [4, 20, 58, 137] X: [2, 5, 21, 24, 89, 108] Z: [5, 21, 59, 138] X: [3, 6, 22, 25, 90, 109] Z: [6, 22, 60, 139] X: [4, 7, 23, 26, 91, 110] Z: [7, 23, 61, 140] X: [5, 8, 24, 27, 92, 111] Z: [8, 24, 62, 141] X: [6, 9, 25, 28, 93, 112] Z: [9, 25, 63, 142] X: [7, 10, 26, 29, 94, 113] Z: [10, 26, 64, 143] X: [8, 11, 27, 30, 95, 114] Z: [11, 27, 65, 144] X: [9, 12, 28, 31, 96, 115] Z: [12, 28, 66, 145] X: [10, 13, 29, 32, 97, 116] Z: [13, 29, 67, 146] X: [11, 14, 30, 33, 98, 117] Z: [14, 30, 68, 147] X: [12, 15, 31, 34, 99, 118] Z: [15, 31, 69, 148] X: [13, 16, 32, 35, 100, 119] Z: [16, 32, 70, 149] X: [14, 17, 33, 36, 101, 120] Z: [17, 33, 71, 150] X: [15, 18, 34, 37, 102, 121] Z: [18, 34, 72, 151] X: [16, 19, 35, 38, 103, 122] Z: [19, 35, 73, 152] X: [17, 20, 36, 39, 104, 123] Z: [20, 36, 74, 153] X: [18, 21, 37, 40, 105, 124] Z: [21, 37, 75, 154] X: [19, 22, 38, 41, 106, 125] Z: [22, 38, 76, 155] X: [20, 23, 39, 42, 107, 126] Z: [23, 39, 77, 156] X: [21, 24, 40, 43, 108, 127] Z: [24, 40, 78, 157] X: [22, 25, 41, 44, 109, 128] Z: [25, 41, 79, 158] X: [23, 26, 42, 45, 110, 129] Z: [26, 42, 80, 159] X: [24, 27, 43, 46, 111, 130] Z: [27, 43, 81, 160] X: [25, 28, 44, 47, 112, 131] Z: [28, 44, 82, 161] X: [26, 29, 45, 48, 113, 132] Z: [29, 45, 83, 162] X: [27, 30, 46, 49, 114, 133] Z: [30, 46, 84, 163] X: [28, 31, 47, 50, 115, 134] Z: [31, 47, 85, 164] X: [29, 32, 48, 51, 116, 135] Z: [32, 48, 86, 165] X: [30, 33, 49, 52, 117, 136] Z: [33, 49, 87, 166] X: [31, 34, 50, 53, 118, 137] Z: [34, 50, 88, 167] X: [32, 35, 51, 54, 119, 138] Z: [35, 51, 89, 168] X: [33, 36, 52, 55, 120, 139] Z: [36, 52, 90, 169] X: [34, 37, 53, 56, 121, 140] Z: [37, 53, 91, 170] X: [35, 38, 54, 57, 122, 141] Z: [0, 38, 54, 92] X: [36, 39, 55, 58, 123, 142] Z: [1, 39, 55, 93] X: [37, 40, 56, 59, 124, 143] Z: [2, 40, 56, 94] X: [38, 41, 57, 60, 125, 144] Z: [3, 41, 57, 95] X: [39, 42, 58, 61, 126, 145] Z: [4, 42, 58, 96] X: [40, 43, 59, 62, 127, 146] Z: [5, 43, 59, 97] X: [41, 44, 60, 63, 128, 147] Z: [6, 44, 60, 98] X: [42, 45, 61, 64, 129, 148] Z: [7, 45, 61, 99] X: [43, 46, 62, 65, 130, 149] Z: [8, 46, 62, 100] X: [44, 47, 63, 66, 131, 150] Z: [9, 47, 63, 101] X: [45, 48, 64, 67, 132, 151] Z: [10, 48, 64, 102] X: [46, 49, 65, 68, 133, 152] Z: [11, 49, 65, 103] X: [47, 50, 66, 69, 134, 153] Z: [12, 50, 66, 104] X: [48, 51, 67, 70, 135, 154] Z: [13, 51, 67, 105] X: [49, 52, 68, 71, 136, 155] Z: [14, 52, 68, 106] X: [50, 53, 69, 72, 137, 156] Z: [15, 53, 69, 107] X: [51, 54, 70, 73, 138, 157] Z: [16, 54, 70, 108] X: [52, 55, 71, 74, 139, 158] Z: [17, 55, 71, 109] X: [53, 56, 72, 75, 140, 159] Z: [18, 56, 72, 110] X: [54, 57, 73, 76, 141, 160] Z: [19, 57, 73, 111] X: [55, 58, 74, 77, 142, 161] Z: [20, 58, 74, 112] X: [56, 59, 75, 78, 143, 162] Z: [21, 59, 75, 113] X: [57, 60, 76, 79, 144, 163] Z: [22, 60, 76, 114] X: [58, 61, 77, 80, 145, 164] Z: [23, 61, 77, 115] X: [59, 62, 78, 81, 146, 165] Z: [24, 62, 78, 116] X: [60, 63, 79, 82, 147, 166] Z: [25, 63, 79, 117] X: [61, 64, 80, 83, 148, 167] Z: [26, 64, 80, 118] X: [62, 65, 81, 84, 149, 168] Z: [27, 65, 81, 119] X: [63, 66, 82, 85, 150, 169] Z: [28, 66, 82, 120] X: [64, 67, 83, 86, 151, 170] Z: [29, 67, 83, 121] X: [0, 65, 68, 84, 87, 152] Z: [30, 68, 84, 122] X: [1, 66, 69, 85, 88, 153] Z: [31, 69, 85, 123] X: [2, 67, 70, 86, 89, 154] Z: [32, 70, 86, 124] X: [3, 68, 71, 87, 90, 155] Z: [33, 71, 87, 125] X: [4, 69, 72, 88, 91, 156] Z: [34, 72, 88, 126] X: [5, 70, 73, 89, 92, 157] Z: [35, 73, 89, 127] X: [6, 71, 74, 90, 93, 158] Z: [36, 74, 90, 128] X: [7, 72, 75, 91, 94, 159] Z: [37, 75, 91, 129] X: [8, 73, 76, 92, 95, 160] Z: [38, 76, 92, 130] X: [9, 74, 77, 93, 96, 161] Z: [39, 77, 93, 131] X: [10, 75, 78, 94, 97, 162] Z: [40, 78, 94, 132] X: [11, 76, 79, 95, 98, 163] Z: [41, 79, 95, 133] X: [12, 77, 80, 96, 99, 164] Z: [42, 80, 96, 134] X: [13, 78, 81, 97, 100, 165] Z: [43, 81, 97, 135] X: [14, 79, 82, 98, 101, 166] Z: [44, 82, 98, 136] X: [15, 80, 83, 99, 102, 167] Z: [45, 83, 99, 137] X: [16, 81, 84, 100, 103, 168] Z: [46, 84, 100, 138] X: [17, 82, 85, 101, 104, 169] Z: [47, 85, 101, 139] X: [18, 83, 86, 102, 105, 170] Z: [48, 86, 102, 140]

Code ID 171-25-7 · download JSON · raw on GitHub