[[91,13,7]] · QEC Challenge

[[91,13,7]] 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)

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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 (2,8)×91 (3,8)×91 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 91 (2,8): 91 (2,10): 91 (2,12): 819 (2,14): 273 (3,8): 91 (3,10): 273 (3,12): 2457 (3,14): 3458 (3,16): 7462 (3,18): 5824 (3,20): 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) = 1 + x¹³ + x⁴⁵ + x⁴⁶ + x⁵⁸ + x⁵⁹, b(x) = x²⁰ + x⁴⁶ + x⁵⁸ + x⁸⁴ 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

[[91,13,7]] — 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 = 91; a(x) = 1 + x^13 + x^45 + x^46 + x^58 + x^59; b(x) = x^20 + x^46 + x^58 + x^84. 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 [], Z on [4, 17, 30, 43, 56, 69, 82].

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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IIIIIIIIIIIIXIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIXYIIIIIIIIIIIYXIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIX

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

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

Code ID 91-13-7 · download JSON · raw on GitHub