[[87,3,14]] · QEC Challenge

[[87,3,14]] d ≤stabilizer

kd²/n

6.759

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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 (claimed upper_bound)

witness operator (Pauli string, 14 qubits)

IIIZIIZIIIIIIXIXIIIIIIZIIZIIZIIIIIIIIIIIIIIIIZIIZIIZIIIIIIXIXIIIIIIZIIZIIIIIIIIIIIIIIII X: [13, 15, 58, 60] Z: [3, 6, 22, 25, 28, 45, 48, 51, 67, 70]

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)×87 (2,12)×1044 (3,12)×377 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 87 (2,12): 1044 (2,14): 348 (3,12): 377 (3,14): 696 (3,16): 15486 (3,18): 11310 (3,20): 1479

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 87 cyclic shifts, a(x) = x⁴¹ + x⁴³ + x⁴⁴ + x⁴⁶, b(x) = x¹⁴ + x³⁷ + x⁵⁰ + x⁷³ in F_2[x]/(x⁸⁷ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁸⁷ - 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_87, 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

[[87,3,14]] — 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 <= 14 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 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 = 87; a(x) = x^41 + x^43 + x^44 + x^46; b(x) = x^14 + x^37 + x^50 + x^73. 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) = 3. Witness: X on [13, 15, 58, 60], Z on [3, 6, 22, 25, 28, 45, 48, 51, 67, 70].

Stabilizer generators

generators 87 (max weight 8; 87 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 (87, Pauli strings on 87 qubits)

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symplectic rows (A | B) (87, sparse supports)

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

Code ID 87-3-14 · download JSON · raw on GitHub