[[66,6,9]] · QEC Challenge

[[66,6,9]] d ≤stabilizer

kd²/n

7.364

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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 9 · witness Pauli weight 9 (claimed upper_bound)

witness operator (Pauli string, 9 qubits)

XIXIIIIIIIZIIZIIZIIIIIIIXIXIIIIIIIIIIIIIIIIIIXIXIIIIIIIIIIIIIIIIII X: [0, 2, 24, 26, 45, 47] Z: [10, 13, 16]

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)×66 (2,8)×66 (3,10)×198 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 66 (2,8): 66 (2,10): 132 (2,12): 528 (2,14): 132 (3,10): 198 (3,12): 1122 (3,14): 4092 (3,16): 5544 (3,18): 3036 (3,20): 330

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 66 cyclic shifts, a(x) = x²⁹ + x³¹ + x³⁵ + x³⁷, b(x) = x¹⁸ + x²⁷ + x³⁹ + x⁴⁸ in F_2[x]/(x⁶⁶ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁶⁶ - 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_66, 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

[[66,6,9]] — 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 <= 9 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 9.

Claim: d <= 9, 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 = 66; a(x) = x^29 + x^31 + x^35 + x^37; b(x) = x^18 + x^27 + x^39 + x^48. 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) = 6. Witness: X on [0, 2, 24, 26, 45, 47], Z on [10, 13, 16].

Stabilizer generators

generators 66 (max weight 8; 66 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 (66, Pauli strings on 66 qubits)

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symplectic rows (A | B) (66, sparse supports)

X: [29, 31, 35, 37] Z: [18, 27, 39, 48] X: [30, 32, 36, 38] Z: [19, 28, 40, 49] X: [31, 33, 37, 39] Z: [20, 29, 41, 50] X: [32, 34, 38, 40] Z: [21, 30, 42, 51] X: [33, 35, 39, 41] Z: [22, 31, 43, 52] X: [34, 36, 40, 42] Z: [23, 32, 44, 53] X: [35, 37, 41, 43] Z: [24, 33, 45, 54] X: [36, 38, 42, 44] Z: [25, 34, 46, 55] X: [37, 39, 43, 45] Z: [26, 35, 47, 56] X: [38, 40, 44, 46] Z: [27, 36, 48, 57] X: [39, 41, 45, 47] Z: [28, 37, 49, 58] X: [40, 42, 46, 48] Z: [29, 38, 50, 59] X: [41, 43, 47, 49] Z: [30, 39, 51, 60] X: [42, 44, 48, 50] Z: [31, 40, 52, 61] X: [43, 45, 49, 51] Z: [32, 41, 53, 62] X: [44, 46, 50, 52] Z: [33, 42, 54, 63] X: [45, 47, 51, 53] Z: [34, 43, 55, 64] X: [46, 48, 52, 54] Z: [35, 44, 56, 65] X: [47, 49, 53, 55] Z: [0, 36, 45, 57] X: [48, 50, 54, 56] Z: [1, 37, 46, 58] X: [49, 51, 55, 57] Z: [2, 38, 47, 59] X: [50, 52, 56, 58] Z: [3, 39, 48, 60] X: [51, 53, 57, 59] Z: [4, 40, 49, 61] X: [52, 54, 58, 60] Z: [5, 41, 50, 62] X: [53, 55, 59, 61] Z: [6, 42, 51, 63] X: [54, 56, 60, 62] Z: [7, 43, 52, 64] X: [55, 57, 61, 63] Z: [8, 44, 53, 65] X: [56, 58, 62, 64] Z: [0, 9, 45, 54] X: [57, 59, 63, 65] Z: [1, 10, 46, 55] X: [0, 58, 60, 64] Z: [2, 11, 47, 56] X: [1, 59, 61, 65] Z: [3, 12, 48, 57] X: [0, 2, 60, 62] Z: [4, 13, 49, 58] X: [1, 3, 61, 63] Z: [5, 14, 50, 59] X: [2, 4, 62, 64] Z: [6, 15, 51, 60] X: [3, 5, 63, 65] Z: [7, 16, 52, 61] X: [0, 4, 6, 64] Z: [8, 17, 53, 62] X: [1, 5, 7, 65] Z: [9, 18, 54, 63] X: [0, 2, 6, 8] Z: [10, 19, 55, 64] X: [1, 3, 7, 9] Z: [11, 20, 56, 65] X: [2, 4, 8, 10] Z: [0, 12, 21, 57] X: [3, 5, 9, 11] Z: [1, 13, 22, 58] X: [4, 6, 10, 12] Z: [2, 14, 23, 59] X: [5, 7, 11, 13] Z: [3, 15, 24, 60] X: [6, 8, 12, 14] Z: [4, 16, 25, 61] X: [7, 9, 13, 15] Z: [5, 17, 26, 62] X: [8, 10, 14, 16] Z: [6, 18, 27, 63] X: [9, 11, 15, 17] Z: [7, 19, 28, 64] X: [10, 12, 16, 18] Z: [8, 20, 29, 65] X: [11, 13, 17, 19] Z: [0, 9, 21, 30] X: [12, 14, 18, 20] Z: [1, 10, 22, 31] X: [13, 15, 19, 21] Z: [2, 11, 23, 32] X: [14, 16, 20, 22] Z: [3, 12, 24, 33] X: [15, 17, 21, 23] Z: [4, 13, 25, 34] X: [16, 18, 22, 24] Z: [5, 14, 26, 35] X: [17, 19, 23, 25] Z: [6, 15, 27, 36] X: [18, 20, 24, 26] Z: [7, 16, 28, 37] X: [19, 21, 25, 27] Z: [8, 17, 29, 38] X: [20, 22, 26, 28] Z: [9, 18, 30, 39] X: [21, 23, 27, 29] Z: [10, 19, 31, 40] X: [22, 24, 28, 30] Z: [11, 20, 32, 41] X: [23, 25, 29, 31] Z: [12, 21, 33, 42] X: [24, 26, 30, 32] Z: [13, 22, 34, 43] X: [25, 27, 31, 33] Z: [14, 23, 35, 44] X: [26, 28, 32, 34] Z: [15, 24, 36, 45] X: [27, 29, 33, 35] Z: [16, 25, 37, 46] X: [28, 30, 34, 36] Z: [17, 26, 38, 47]

Code ID 66-6-9 · download JSON · raw on GitHub