[[169,13,8]] · QEC Challenge

[[169,13,8]] d ≤stabilizer

169

kd²/n

4.923

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 (2 Y factors; Hamming weight over 2n bits 10) (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 6

qubit degrees S 6

trapping sets S (1,6)×169 (2,8)×1014 (3,10)×6422 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 169 (2,8): 1014 (2,10): 507 (3,10): 6422 (3,12): 10140 (3,14): 2535

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 169 cyclic shifts, a(x) = x⁷⁷ + x⁷⁹ + x⁹⁰ + x⁹², b(x) = x⁴⁰ + x⁷⁹ + x⁹⁰ + x¹²⁹ in F_2[x]/(x¹⁶⁹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁶⁹ - 1)

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:quant-ph/0502086

date 2026-10-01

notes Found by a sweep of one-block palindromic cyclic codes (Camara-Ollivier-Tillich-type group-algebra code over Z_169, 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 ≤ 6 (computed)

How this code was found

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

[[169,13,8]] — one-block palindromic cyclic stabilizer code X^a Z^b, weight 6

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 = 169; a(x) = x^77 + x^79 + x^90 + x^92; b(x) = x^40 + x^79 + x^90 + x^129. 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) = 13. Witness: X on [22, 24, 74, 76, 113, 115, 152, 154], Z on [24, 74].

Stabilizer generators

generators 169 (max weight 6; 169 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 (169, Pauli strings on 169 qubits)

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symplectic rows (A | B) (169, sparse supports)

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

Code ID 169-13-8 · download JSON · raw on GitHub