[[119,23,6]] · QEC Challenge

[[119,23,6]] d ≤stabilizer

119

kd²/n

6.958

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 6 · witness Pauli weight 6 (claimed upper_bound)

witness operator (Pauli string, 6 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)×119 (2,12)×1428 (3,12)×595 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 119 (2,12): 1428 (2,14): 476 (3,12): 595 (3,14): 952 (3,16): 20111 (3,18): 17136 (3,20): 2856

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 119 cyclic shifts, a(x) = x⁴⁴ + x⁵⁸ + x⁶¹ + x⁷⁵, b(x) = x³² + x⁵³ + x⁶⁶ + x⁸⁷ in F_2[x]/(x¹¹⁹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹¹⁹ - 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_119, 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

[[119,23,6]] — 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 <= 6 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 6.

Claim: d <= 6, 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 = 119; a(x) = x^44 + x^58 + x^61 + x^75; b(x) = x^32 + x^53 + x^66 + x^87. 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) = 23. Witness: X on [18, 32], Z on [6, 23, 27, 44].

Stabilizer generators

generators 119 (max weight 8; 119 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 (119, Pauli strings on 119 qubits)

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symplectic rows (A | B) (119, sparse supports)

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

Code ID 119-23-6 · download JSON · raw on GitHub