[[91,7,8]] · QEC Challenge

[[91,7,8]] d ≤stabilizer

kd²/n

4.923

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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 8 · witness Pauli weight 8 (claimed upper_bound)

witness operator (Pauli string, 8 qubits)

IIIIXIXIIIIIIIIIIIIIIIIIIIIIIIIIXIXIIIIZIIIIIIIIIIIIIIIIIIIIIIZIIIIXIXIIIIIIIIIIIIIIIIIIIII X: [4, 6, 32, 34, 67, 69] Z: [39, 62]

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 (3,10)×3458 (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 (3,10): 3458 (3,12): 5460 (3,14): 1365

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) = x⁴¹ + x⁴³ + x⁴⁸ + x⁵⁰, b(x) = x²⁰ + x⁴³ + x⁴⁸ + x⁷¹ 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

[[91,7,8]] — 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 <= 8 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 8.

Claim: d <= 8, 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^41 + x^43 + x^48 + x^50; b(x) = x^20 + x^43 + x^48 + x^71. 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) = 7. Witness: X on [4, 6, 32, 34, 67, 69], Z on [39, 62].

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

Code ID 91-7-8 · download JSON · raw on GitHub