[[91,13,6]] · QEC Challenge

[[91,13,6]] d ≤stabilizer

kd²/n

5.143

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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)

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIZIIIIIIIIIIXIXIIIIIIIIIIZIIIIIIIIIIIIZIIIIIII X: [57, 59] Z: [33, 46, 70, 83]

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 6

qubit degrees S 6

trapping sets S (1,6)×91 (2,8)×546 (3,6)×273 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 91 (2,8): 546 (2,10): 273 (3,6): 273 (3,8): 546 (3,10): 2639 (3,12): 3276 (3,14): 819

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) = x³⁸ + x⁴⁰ + x⁵¹ + x⁵³, b(x) = x¹⁴ + x⁴⁰ + x⁵¹ + x⁷⁷ 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 2026-10-01

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[91,13,6]] — one-block palindromic cyclic stabilizer code X^a Z^b, weight 6

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^38 + x^40 + x^51 + x^53; b(x) = x^14 + x^40 + x^51 + x^77. 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 [57, 59], Z on [33, 46, 70, 83].

Stabilizer generators

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

Code ID 91-13-6 · download JSON · raw on GitHub