[[63,9,7]] · QEC Challenge

[[63,9,7]] d ≤stabilizer

kd²/n

7.0

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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 7 · witness Pauli weight 7 (claimed upper_bound)

witness operator (Pauli string, 7 qubits)

IIIIIIIZIIIIIIIXXIIIIIIIZIIIIIIIIZIIIIIIIIIIIIIXIIIIIIIIIIIIIZI X: [15, 16, 47] Z: [7, 24, 33, 61]

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): 189 (3,12): 1701 (3,14): 2394 (3,16): 5166 (3,18): 4032 (3,20): 567

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 + x⁹ + x³¹ + x³² + x⁴⁰ + x⁴¹, b(x) = x¹⁴ + x³² + x⁴⁰ + x⁵⁸ 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

[[63,9,7]] — 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 = 63; a(x) = 1 + x^9 + x^31 + x^32 + x^40 + x^41; b(x) = x^14 + x^32 + x^40 + x^58. 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) = 9. Witness: X on [15, 16, 47], Z on [7, 24, 33, 61].

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)

XIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIII IXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIII IIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZII IIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZI IIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZ ZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIII IZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIII IIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIII IIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIII IIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIII IIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIII IIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIII IIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIII IIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIII IIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIII IIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIII IIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIII IIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIII IIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIII IIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXII IIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXI IIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYX XIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIY YXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIII IYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIII IIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIII IIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIII IIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIII IIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYII IIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYI IIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXY YIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIIIX XYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIIII IXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIIII IIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIIII IIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIIII IIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIIII IIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIIII IIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIIII IIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIIII IIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIIII IIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIIII IIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIIII IIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIIII IIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIIII IIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZIII IIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZII IIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZI IIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIIIZ ZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIIII IZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXIII IIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXII IIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIXI IIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIIIX XIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIIII IXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIIII IIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIIII IIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIIII IIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIIII IIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXIII IIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXII IIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIXI IIIIIIIIXIIIIZIIIIIIIIIIIIIIIIXYIIIIIIIYXIIIIIIIIIIIIIIIIZIIIIX

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

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

Code ID 63-9-7 · download JSON · raw on GitHub