[[117,19,7]] · QEC Challenge

[[117,19,7]] d ≤stabilizer

117

kd²/n

7.957

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)×117 (2,8)×117 (3,8)×117 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 117 (2,8): 117 (2,10): 117 (2,12): 1053 (2,14): 351 (3,8): 117 (3,10): 234 (3,12): 2847 (3,14): 3744 (3,16): 11817 (3,18): 8892 (3,20): 1287

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 117 cyclic shifts, a(x) = 1 + x¹³ + x⁵⁷ + x⁶⁰ + x⁷⁰ + x⁷³, b(x) = x³⁴ + x⁶⁰ + x⁷⁰ + x⁹⁶ in F_2[x]/(x¹¹⁷ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹¹⁷ - 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_117, 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

[[117,19,7]] — 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 = 117; a(x) = 1 + x^13 + x^57 + x^60 + x^70 + x^73; b(x) = x^34 + x^60 + x^70 + x^96. 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) = 19. Witness: X on [47, 50, 107], Z on [24, 37, 60, 73].

Stabilizer generators

generators 117 (max weight 8; 117 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 (117, Pauli strings on 117 qubits)

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IIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXIIIIIII IIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXIIIIII IIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXIIIII IIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXIIII IIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXIII IIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXII IIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIXI IIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIX XIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIII IXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIII IIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIIII IIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIIII IIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIIII IIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIIII IIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIIII IIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIIII IIIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIIII IIIIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXIII IIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXII IIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIXI IIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIYIIXIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIX

symplectic rows (A | B) (117, sparse supports)

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

Code ID 117-19-7 · download JSON · raw on GitHub