[[94,2,14]] · QEC Challenge

[[94,2,14]] d ≤stabilizer

kd²/n

4.17

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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 14 · witness Pauli weight 14 (10 Y factors; Hamming weight over 2n bits 24) (claimed upper_bound)

witness operator (Pauli string, 14 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 6

qubit degrees S 6

trapping sets S (1,6)×94 (2,8)×564 (3,10)×3572 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 94 (2,8): 564 (2,10): 282 (3,10): 3572 (3,12): 5640 (3,14): 1410

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 94 cyclic shifts, a(x) = x⁴⁰ + x⁴⁴ + x⁵⁰ + x⁵⁴, b(x) = x³¹ + x⁴⁰ + x⁵⁴ + x⁶³ in F_2[x]/(x⁹⁴ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁹⁴ - 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_94, 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 (dropped 18->14). 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

[[94,2,14]] — 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 <= 14 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 18->14).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 14.

Claim: d <= 14, 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 = 94; a(x) = x^40 + x^44 + x^50 + x^54; b(x) = x^31 + x^40 + x^54 + x^63. 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) = 2. Witness: X on [5, 10, 15, 28, 29, 38, 47, 48, 57, 66, 67, 76, 85, 86], Z on [10, 29, 38, 47, 48, 57, 66, 67, 76, 85].

Stabilizer generators

generators 94 (max weight 6; 94 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 (94, Pauli strings on 94 qubits)

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symplectic rows (A | B) (94, sparse supports)

X: [40, 44, 50, 54] Z: [31, 40, 54, 63] X: [41, 45, 51, 55] Z: [32, 41, 55, 64] X: [42, 46, 52, 56] Z: [33, 42, 56, 65] X: [43, 47, 53, 57] Z: [34, 43, 57, 66] X: [44, 48, 54, 58] Z: [35, 44, 58, 67] X: [45, 49, 55, 59] Z: [36, 45, 59, 68] X: [46, 50, 56, 60] Z: [37, 46, 60, 69] X: [47, 51, 57, 61] Z: [38, 47, 61, 70] X: [48, 52, 58, 62] Z: [39, 48, 62, 71] X: [49, 53, 59, 63] Z: [40, 49, 63, 72] X: [50, 54, 60, 64] Z: [41, 50, 64, 73] X: [51, 55, 61, 65] Z: [42, 51, 65, 74] X: [52, 56, 62, 66] Z: [43, 52, 66, 75] X: [53, 57, 63, 67] Z: [44, 53, 67, 76] X: [54, 58, 64, 68] Z: [45, 54, 68, 77] X: [55, 59, 65, 69] Z: [46, 55, 69, 78] X: [56, 60, 66, 70] Z: [47, 56, 70, 79] X: [57, 61, 67, 71] Z: [48, 57, 71, 80] X: [58, 62, 68, 72] Z: [49, 58, 72, 81] X: [59, 63, 69, 73] Z: [50, 59, 73, 82] X: [60, 64, 70, 74] Z: [51, 60, 74, 83] X: [61, 65, 71, 75] Z: [52, 61, 75, 84] X: [62, 66, 72, 76] Z: [53, 62, 76, 85] X: [63, 67, 73, 77] Z: [54, 63, 77, 86] X: [64, 68, 74, 78] Z: [55, 64, 78, 87] X: [65, 69, 75, 79] Z: [56, 65, 79, 88] X: [66, 70, 76, 80] Z: [57, 66, 80, 89] X: [67, 71, 77, 81] Z: [58, 67, 81, 90] X: [68, 72, 78, 82] Z: [59, 68, 82, 91] X: [69, 73, 79, 83] Z: [60, 69, 83, 92] X: [70, 74, 80, 84] Z: [61, 70, 84, 93] X: [71, 75, 81, 85] Z: [0, 62, 71, 85] X: [72, 76, 82, 86] Z: [1, 63, 72, 86] X: [73, 77, 83, 87] Z: [2, 64, 73, 87] X: [74, 78, 84, 88] Z: [3, 65, 74, 88] X: [75, 79, 85, 89] Z: [4, 66, 75, 89] X: [76, 80, 86, 90] Z: [5, 67, 76, 90] X: [77, 81, 87, 91] Z: [6, 68, 77, 91] X: [78, 82, 88, 92] Z: [7, 69, 78, 92] X: [79, 83, 89, 93] Z: [8, 70, 79, 93] X: [0, 80, 84, 90] Z: [0, 9, 71, 80] X: [1, 81, 85, 91] Z: [1, 10, 72, 81] X: [2, 82, 86, 92] Z: [2, 11, 73, 82] X: [3, 83, 87, 93] Z: [3, 12, 74, 83] X: [0, 4, 84, 88] Z: [4, 13, 75, 84] X: [1, 5, 85, 89] Z: [5, 14, 76, 85] X: [2, 6, 86, 90] Z: [6, 15, 77, 86] X: [3, 7, 87, 91] Z: [7, 16, 78, 87] X: [4, 8, 88, 92] Z: [8, 17, 79, 88] X: [5, 9, 89, 93] Z: [9, 18, 80, 89] X: [0, 6, 10, 90] Z: [10, 19, 81, 90] X: [1, 7, 11, 91] Z: [11, 20, 82, 91] X: [2, 8, 12, 92] Z: [12, 21, 83, 92] X: [3, 9, 13, 93] Z: [13, 22, 84, 93] X: [0, 4, 10, 14] Z: [0, 14, 23, 85] X: [1, 5, 11, 15] Z: [1, 15, 24, 86] X: [2, 6, 12, 16] Z: [2, 16, 25, 87] X: [3, 7, 13, 17] Z: [3, 17, 26, 88] X: [4, 8, 14, 18] Z: [4, 18, 27, 89] X: [5, 9, 15, 19] Z: [5, 19, 28, 90] X: [6, 10, 16, 20] Z: [6, 20, 29, 91] X: [7, 11, 17, 21] Z: [7, 21, 30, 92] X: [8, 12, 18, 22] Z: [8, 22, 31, 93] X: [9, 13, 19, 23] Z: [0, 9, 23, 32] X: [10, 14, 20, 24] Z: [1, 10, 24, 33] X: [11, 15, 21, 25] Z: [2, 11, 25, 34] X: [12, 16, 22, 26] Z: [3, 12, 26, 35] X: [13, 17, 23, 27] Z: [4, 13, 27, 36] X: [14, 18, 24, 28] Z: [5, 14, 28, 37] X: [15, 19, 25, 29] Z: [6, 15, 29, 38] X: [16, 20, 26, 30] Z: [7, 16, 30, 39] X: [17, 21, 27, 31] Z: [8, 17, 31, 40] X: [18, 22, 28, 32] Z: [9, 18, 32, 41] X: [19, 23, 29, 33] Z: [10, 19, 33, 42] X: [20, 24, 30, 34] Z: [11, 20, 34, 43] X: [21, 25, 31, 35] Z: [12, 21, 35, 44] X: [22, 26, 32, 36] Z: [13, 22, 36, 45] X: [23, 27, 33, 37] Z: [14, 23, 37, 46] X: [24, 28, 34, 38] Z: [15, 24, 38, 47] X: [25, 29, 35, 39] Z: [16, 25, 39, 48] X: [26, 30, 36, 40] Z: [17, 26, 40, 49] X: [27, 31, 37, 41] Z: [18, 27, 41, 50] X: [28, 32, 38, 42] Z: [19, 28, 42, 51] X: [29, 33, 39, 43] Z: [20, 29, 43, 52] X: [30, 34, 40, 44] Z: [21, 30, 44, 53] X: [31, 35, 41, 45] Z: [22, 31, 45, 54] X: [32, 36, 42, 46] Z: [23, 32, 46, 55] X: [33, 37, 43, 47] Z: [24, 33, 47, 56] X: [34, 38, 44, 48] Z: [25, 34, 48, 57] X: [35, 39, 45, 49] Z: [26, 35, 49, 58] X: [36, 40, 46, 50] Z: [27, 36, 50, 59] X: [37, 41, 47, 51] Z: [28, 37, 51, 60] X: [38, 42, 48, 52] Z: [29, 38, 52, 61] X: [39, 43, 49, 53] Z: [30, 39, 53, 62]

Code ID 94-2-14 · download JSON · raw on GitHub