[[63,13,6]] · QEC Challenge

[[63,13,6]] d ≤stabilizer

kd²/n

7.429

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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 6 · witness Pauli weight 6 (claimed upper_bound)

witness operator (Pauli string, 6 qubits)

ZIIIIIIZIIIXIIIIIIIIIIIIIIIIIXIIIZIIIIIIZIIIIIIIIIIIIIIIIIIIIII X: [11, 29] Z: [0, 7, 33, 40]

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)×63 (2,8)×63 (3,8)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 63 (2,8): 63 (2,10): 63 (2,12): 567 (2,14): 189 (3,8): 63 (3,10): 126 (3,12): 1533 (3,14): 2016 (3,16): 6363 (3,18): 4788 (3,20): 693

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 63 cyclic shifts, a(x) = 1 + x⁷ + x³⁰ + x³³ + x³⁷ + x⁴⁰, b(x) = x¹⁹ + x³³ + x³⁷ + x⁵¹ in F_2[x]/(x⁶³ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁶³ - 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_63, 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

[[63,13,6]] — 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 = 63; a(x) = 1 + x^7 + x^30 + x^33 + x^37 + x^40; b(x) = x^19 + x^33 + x^37 + x^51. 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) = 13. Witness: X on [11, 29], Z on [0, 7, 33, 40].

Stabilizer generators

generators 63 (max weight 8; 63 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 (63, Pauli strings on 63 qubits)

XIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIII IXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIII IIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIII IIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIII IIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIII IIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIII IIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIII IIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIII IIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIII IIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZII IIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZI IIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZ ZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIII IZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIII IIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIII IIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIII IIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIII IIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIII IIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIII IIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIII IIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXII IIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXI IIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIX XIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYII IXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYI IIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIY YIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIII IYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYII IIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYI IIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIY YIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXII IYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIXI IIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIIIX XIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIIII IXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIIII IIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIIII IIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIIII IIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIIII IIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIIII IIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIIII IIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZIII IIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZII IIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZI IIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIIIZ ZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIIII IZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIIII IIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIIII IIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIIII IIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIIII IIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIIII IIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIIII IIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIIII IIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXIII IIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXII IIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIXI IIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIIIX XIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIIII IXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIIII IIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIIII IIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXIII IIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXII IIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIXI IIIIIIXIIIIIIIIIIIZIIIIIIIIIIXIIYIIIYIIXIIIIIIIIIIZIIIIIIIIIIIX

symplectic rows (A | B) (63, sparse supports)

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

Code ID 63-13-6 · download JSON · raw on GitHub