[[137,1,19]] · QEC Challenge

[[137,1,19]] d ≤stabilizer

137

kd²/n

2.635

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 19 · witness Pauli weight 19 (5 Y factors; Hamming weight over 2n bits 24) (claimed upper_bound)

witness operator (Pauli string, 19 qubits)

IIIZIXIIXIIIIIIIIXIIIIIIIIIIIIIIIIYIIIIIIIIIIIIXIIXIZIYIXIIXIIIIIIIIIIIIIIYIIIIIIIIIIIIIIZIXIIIIIIXIIXIZIIIIIIIIIIYIIIIIIIIIIIIIIIIYIIIII X: [5, 8, 17, 34, 47, 50, 54, 56, 59, 74, 91, 98, 101, 114, 131] Z: [3, 34, 52, 54, 74, 89, 103, 114, 131]

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)×137 (2,8)×822 (3,10)×5343 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 137 (2,8): 822 (2,10): 411 (3,10): 5343 (3,12): 7946 (3,14): 1918

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 137 cyclic shifts, a(x) = x⁶⁶ + x⁶⁸ + x⁶⁹ + x⁷¹, b(x) = x²⁹ + x⁶⁶ + x⁷¹ + x¹⁰⁸ in F_2[x]/(x¹³⁷ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹³⁷ - 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_137, 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 25->19). 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

[[137,1,19]] — 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 <= 19 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 25->19).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 19.

Claim: d <= 19, 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 = 137; a(x) = x^66 + x^68 + x^69 + x^71; b(x) = x^29 + x^66 + x^71 + x^108. 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) = 1. Witness: X on [5, 8, 17, 34, 47, 50, 54, 56, 59, 74, 91, 98, 101, 114, 131], Z on [3, 34, 52, 54, 74, 89, 103, 114, 131].

Stabilizer generators

generators 137 (max weight 6; 137 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 (137, Pauli strings on 137 qubits)

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symplectic rows (A | B) (137, sparse supports)

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

Code ID 137-1-19 · download JSON · raw on GitHub