[[105,25,5]] · QEC Challenge

[[105,25,5]] d ≤stabilizer

105

kd²/n

5.952

Share this result

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 5 · witness Pauli weight 5 (claimed upper_bound)

witness operator (Pauli string, 5 qubits)

IIZIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIII X: [] Z: [2, 23, 44, 65, 86]

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)×105 (2,12)×1260 (3,12)×1050 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 105 (2,12): 1260 (2,14): 420 (3,12): 1050 (3,14): 840 (3,16): 16170 (3,18): 15120 (3,20): 2520

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 105 cyclic shifts, a(x) = x³⁷ + x⁴⁷ + x⁵⁸ + x⁶⁸, b(x) = x²⁴ + x³⁹ + x⁶⁶ + x⁸¹ in F_2[x]/(x¹⁰⁵ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁰⁵ - 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_105, 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

[[105,25,5]] — 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 <= 5 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 5.

Claim: d <= 5, 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 = 105; a(x) = x^37 + x^47 + x^58 + x^68; b(x) = x^24 + x^39 + x^66 + x^81. 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) = 25. Witness: X on [], Z on [2, 23, 44, 65, 86].

Stabilizer generators

generators 105 (max weight 8; 105 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 (105, Pauli strings on 105 qubits)

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symplectic rows (A | B) (105, sparse supports)

X: [37, 47, 58, 68] Z: [24, 39, 66, 81] X: [38, 48, 59, 69] Z: [25, 40, 67, 82] X: [39, 49, 60, 70] Z: [26, 41, 68, 83] X: [40, 50, 61, 71] Z: [27, 42, 69, 84] X: [41, 51, 62, 72] Z: [28, 43, 70, 85] X: [42, 52, 63, 73] Z: [29, 44, 71, 86] X: [43, 53, 64, 74] Z: [30, 45, 72, 87] X: [44, 54, 65, 75] Z: [31, 46, 73, 88] X: [45, 55, 66, 76] Z: [32, 47, 74, 89] X: [46, 56, 67, 77] Z: [33, 48, 75, 90] X: [47, 57, 68, 78] Z: [34, 49, 76, 91] X: [48, 58, 69, 79] Z: [35, 50, 77, 92] X: [49, 59, 70, 80] Z: [36, 51, 78, 93] X: [50, 60, 71, 81] Z: [37, 52, 79, 94] X: [51, 61, 72, 82] Z: [38, 53, 80, 95] X: [52, 62, 73, 83] Z: [39, 54, 81, 96] X: [53, 63, 74, 84] Z: [40, 55, 82, 97] X: [54, 64, 75, 85] Z: [41, 56, 83, 98] X: [55, 65, 76, 86] Z: [42, 57, 84, 99] X: [56, 66, 77, 87] Z: [43, 58, 85, 100] X: [57, 67, 78, 88] Z: [44, 59, 86, 101] X: [58, 68, 79, 89] Z: [45, 60, 87, 102] X: [59, 69, 80, 90] Z: [46, 61, 88, 103] X: [60, 70, 81, 91] Z: [47, 62, 89, 104] X: [61, 71, 82, 92] Z: [0, 48, 63, 90] X: [62, 72, 83, 93] Z: [1, 49, 64, 91] X: [63, 73, 84, 94] Z: [2, 50, 65, 92] X: [64, 74, 85, 95] Z: [3, 51, 66, 93] X: [65, 75, 86, 96] Z: [4, 52, 67, 94] X: [66, 76, 87, 97] Z: [5, 53, 68, 95] X: [67, 77, 88, 98] Z: [6, 54, 69, 96] X: [68, 78, 89, 99] Z: [7, 55, 70, 97] X: [69, 79, 90, 100] Z: [8, 56, 71, 98] X: [70, 80, 91, 101] Z: [9, 57, 72, 99] X: [71, 81, 92, 102] Z: [10, 58, 73, 100] X: [72, 82, 93, 103] Z: [11, 59, 74, 101] X: [73, 83, 94, 104] Z: [12, 60, 75, 102] X: [0, 74, 84, 95] Z: [13, 61, 76, 103] X: [1, 75, 85, 96] Z: [14, 62, 77, 104] X: [2, 76, 86, 97] Z: [0, 15, 63, 78] X: [3, 77, 87, 98] Z: [1, 16, 64, 79] X: [4, 78, 88, 99] Z: [2, 17, 65, 80] X: [5, 79, 89, 100] Z: [3, 18, 66, 81] X: [6, 80, 90, 101] Z: [4, 19, 67, 82] X: [7, 81, 91, 102] Z: [5, 20, 68, 83] X: [8, 82, 92, 103] Z: [6, 21, 69, 84] X: [9, 83, 93, 104] Z: [7, 22, 70, 85] X: [0, 10, 84, 94] Z: [8, 23, 71, 86] X: [1, 11, 85, 95] Z: [9, 24, 72, 87] X: [2, 12, 86, 96] Z: [10, 25, 73, 88] X: [3, 13, 87, 97] Z: [11, 26, 74, 89] X: [4, 14, 88, 98] Z: [12, 27, 75, 90] X: [5, 15, 89, 99] Z: [13, 28, 76, 91] X: [6, 16, 90, 100] Z: [14, 29, 77, 92] X: [7, 17, 91, 101] Z: [15, 30, 78, 93] X: [8, 18, 92, 102] Z: [16, 31, 79, 94] X: [9, 19, 93, 103] Z: [17, 32, 80, 95] X: [10, 20, 94, 104] Z: [18, 33, 81, 96] X: [0, 11, 21, 95] Z: [19, 34, 82, 97] X: [1, 12, 22, 96] Z: [20, 35, 83, 98] X: [2, 13, 23, 97] Z: [21, 36, 84, 99] X: [3, 14, 24, 98] Z: [22, 37, 85, 100] X: [4, 15, 25, 99] Z: [23, 38, 86, 101] X: [5, 16, 26, 100] Z: [24, 39, 87, 102] X: [6, 17, 27, 101] Z: [25, 40, 88, 103] X: [7, 18, 28, 102] Z: [26, 41, 89, 104] X: [8, 19, 29, 103] Z: [0, 27, 42, 90] X: [9, 20, 30, 104] Z: [1, 28, 43, 91] X: [0, 10, 21, 31] Z: [2, 29, 44, 92] X: [1, 11, 22, 32] Z: [3, 30, 45, 93] X: [2, 12, 23, 33] Z: [4, 31, 46, 94] X: [3, 13, 24, 34] Z: [5, 32, 47, 95] X: [4, 14, 25, 35] Z: [6, 33, 48, 96] X: [5, 15, 26, 36] Z: [7, 34, 49, 97] X: [6, 16, 27, 37] Z: [8, 35, 50, 98] X: [7, 17, 28, 38] Z: [9, 36, 51, 99] X: [8, 18, 29, 39] Z: [10, 37, 52, 100] X: [9, 19, 30, 40] Z: [11, 38, 53, 101] X: [10, 20, 31, 41] Z: [12, 39, 54, 102] X: [11, 21, 32, 42] Z: [13, 40, 55, 103] X: [12, 22, 33, 43] Z: [14, 41, 56, 104] X: [13, 23, 34, 44] Z: [0, 15, 42, 57] X: [14, 24, 35, 45] Z: [1, 16, 43, 58] X: [15, 25, 36, 46] Z: [2, 17, 44, 59] X: [16, 26, 37, 47] Z: [3, 18, 45, 60] X: [17, 27, 38, 48] Z: [4, 19, 46, 61] X: [18, 28, 39, 49] Z: [5, 20, 47, 62] X: [19, 29, 40, 50] Z: [6, 21, 48, 63] X: [20, 30, 41, 51] Z: [7, 22, 49, 64] X: [21, 31, 42, 52] Z: [8, 23, 50, 65] X: [22, 32, 43, 53] Z: [9, 24, 51, 66] X: [23, 33, 44, 54] Z: [10, 25, 52, 67] X: [24, 34, 45, 55] Z: [11, 26, 53, 68] X: [25, 35, 46, 56] Z: [12, 27, 54, 69] X: [26, 36, 47, 57] Z: [13, 28, 55, 70] X: [27, 37, 48, 58] Z: [14, 29, 56, 71] X: [28, 38, 49, 59] Z: [15, 30, 57, 72] X: [29, 39, 50, 60] Z: [16, 31, 58, 73] X: [30, 40, 51, 61] Z: [17, 32, 59, 74] X: [31, 41, 52, 62] Z: [18, 33, 60, 75] X: [32, 42, 53, 63] Z: [19, 34, 61, 76] X: [33, 43, 54, 64] Z: [20, 35, 62, 77] X: [34, 44, 55, 65] Z: [21, 36, 63, 78] X: [35, 45, 56, 66] Z: [22, 37, 64, 79] X: [36, 46, 57, 67] Z: [23, 38, 65, 80]

Code ID 105-25-5 · download JSON · raw on GitHub