[[91,19,6]] · QEC Challenge

[[91,19,6]] d ≤stabilizer

kd²/n

7.516

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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)

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIXIIZIIIIIIIZIIXIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII X: [41, 55] Z: [31, 44, 52, 65]

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)×91 (2,12)×1092 (3,12)×455 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 91 (2,12): 1092 (2,14): 364 (3,12): 455 (3,14): 728 (3,16): 15561 (3,18): 12740 (3,20): 2002

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 91 cyclic shifts, a(x) = x³² + x⁴⁵ + x⁴⁶ + x⁵⁹, b(x) = x²² + x⁴³ + x⁴⁸ + x⁶⁹ in F_2[x]/(x⁹¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁹¹ - 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_91, 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

[[91,19,6]] — 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 = 91; a(x) = x^32 + x^45 + x^46 + x^59; b(x) = x^22 + x^43 + x^48 + x^69. 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 [41, 55], Z on [31, 44, 52, 65].

Stabilizer generators

generators 91 (max weight 8; 91 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 (91, Pauli strings on 91 qubits)

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symplectic rows (A | B) (91, sparse supports)

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

Code ID 91-19-6 · download JSON · raw on GitHub