[[105,21,6]] · QEC Challenge

[[105,21,6]] d ≤stabilizer

105

kd²/n

7.2

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 6 · witness Pauli weight 6 (claimed upper_bound)

witness operator (Pauli string, 6 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)×105 (2,12)×1260 (3,12)×735 (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): 735 (3,14): 840 (3,16): 17325 (3,18): 14700 (3,20): 2310

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) = x³⁸ + x⁵² + x⁵³ + x⁶⁷, b(x) = x¹³ + x⁴³ + x⁶² + x⁹² 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

[[105,21,6]] — 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 <= 6 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 6.

Claim: d <= 6, 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^38 + x^52 + x^53 + x^67; b(x) = x^13 + x^43 + x^62 + x^92. 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) = 21. Witness: X on [14, 21, 70, 77], Z on [38, 53].

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

Code ID 105-21-6 · download JSON · raw on GitHub