[[99,17,7]] · QEC Challenge

[[99,17,7]] d ≤stabilizer

kd²/n

8.414

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)

IIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIXIIXIIIIIIIIIIZIIIIIIIIIIZIIIIIIIIIIIIIIIII X: [8, 56, 59] Z: [34, 45, 70, 81]

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)×99 (2,8)×99 (3,8)×99 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 99 (2,8): 99 (2,10): 99 (2,12): 891 (2,14): 297 (3,8): 99 (3,10): 198 (3,12): 2409 (3,14): 3168 (3,16): 9999 (3,18): 7524 (3,20): 1089

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 99 cyclic shifts, a(x) = 1 + x¹¹ + x⁴⁸ + x⁵¹ + x⁵⁹ + x⁶², b(x) = x²⁶ + x⁴⁸ + x⁶² + x⁸⁴ in F_2[x]/(x⁹⁹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁹⁹ - 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_99, 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

[[99,17,7]] — 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 = 99; a(x) = 1 + x^11 + x^48 + x^51 + x^59 + x^62; b(x) = x^26 + x^48 + x^62 + x^84. 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) = 17. Witness: X on [8, 56, 59], Z on [34, 45, 70, 81].

Stabilizer generators

generators 99 (max weight 8; 99 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 (99, Pauli strings on 99 qubits)

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IIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIIIIIIII IIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIIIIIII IIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIIIIII IIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIIIII IIIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIIII IIIIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXIII IIIIIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXII IIIIIIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIXI IIIIIIIIIIXIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIYIIXIIIIIIIXIIYIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIX

symplectic rows (A | B) (99, sparse supports)

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

Code ID 99-17-7 · download JSON · raw on GitHub