[[134,2,17]] · QEC Challenge

[[134,2,17]] d ≤stabilizer

134

kd²/n

4.313

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 17 · witness Pauli weight 17 (3 Y factors; Hamming weight over 2n bits 20) (claimed upper_bound)

witness operator (Pauli string, 17 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)×134 (2,8)×804 (3,10)×5092 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 134 (2,8): 804 (2,10): 402 (3,10): 5092 (3,12): 8040 (3,14): 2010

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 134 cyclic shifts, a(x) = x⁶⁵ + x⁶⁹, b(x) = x⁵ + x²² + x¹¹² + x¹²⁹ in F_2[x]/(x¹³⁴ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹³⁴ - 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_134, 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 (dropped 19->17). 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

[[134,2,17]] — 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 <= 17 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 19->17).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 17.

Claim: d <= 17, 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 = 134; a(x) = x^65 + x^69; b(x) = x^5 + x^22 + x^112 + 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) = 2. Witness: X on [0, 7, 13, 30, 70, 87, 97, 100, 114, 117, 124], Z on [13, 40, 44, 48, 52, 56, 87, 94, 117].

Stabilizer generators

generators 134 (max weight 6; 134 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 (134, Pauli strings on 134 qubits)

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symplectic rows (A | B) (134, sparse supports)

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

Code ID 134-2-17 · download JSON · raw on GitHub