[[111,3,12]] · QEC Challenge

[[111,3,12]] d ≤stabilizer

111

kd²/n

3.892

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 12 · witness Pauli weight 12 (10 Y factors; Hamming weight over 2n bits 22) (claimed upper_bound)

witness operator (Pauli string, 12 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)×111 (2,8)×666 (3,10)×4329 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 111 (2,8): 666 (2,10): 333 (3,10): 4329 (3,12): 6438 (3,14): 1554

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 111 cyclic shifts, a(x) = x⁵³ + x⁵⁵ + x⁵⁶ + x⁵⁸, b(x) = x¹⁰ + x⁵³ + x⁵⁸ + x¹⁰¹ in F_2[x]/(x¹¹¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹¹¹ - 1)

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:quant-ph/0502086

date 2026-10-01

notes Found by a sweep of one-block palindromic cyclic codes (Camara-Ollivier-Tillich-type group-algebra code over Z_111, 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 14->12). 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

[[111,3,12]] — 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 <= 12 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 14->12).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 12.

Claim: d <= 12, 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 = 111; a(x) = x^53 + x^55 + x^56 + x^58; b(x) = x^10 + x^53 + x^58 + x^101. 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 [16, 18, 37, 39, 61, 63, 82, 84, 103, 105], Z on [16, 18, 37, 39, 60, 61, 63, 82, 84, 103, 105, 106].

Stabilizer generators

generators 111 (max weight 6; 111 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 (111, Pauli strings on 111 qubits)

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IIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIII IIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIII IIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIII IIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIII IIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIII IIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIII IIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIXXIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIII

symplectic rows (A | B) (111, sparse supports)

X: [53, 55, 56, 58] Z: [10, 53, 58, 101] X: [54, 56, 57, 59] Z: [11, 54, 59, 102] X: [55, 57, 58, 60] Z: [12, 55, 60, 103] X: [56, 58, 59, 61] Z: [13, 56, 61, 104] X: [57, 59, 60, 62] Z: [14, 57, 62, 105] X: [58, 60, 61, 63] Z: [15, 58, 63, 106] X: [59, 61, 62, 64] Z: [16, 59, 64, 107] X: [60, 62, 63, 65] Z: [17, 60, 65, 108] X: [61, 63, 64, 66] Z: [18, 61, 66, 109] X: [62, 64, 65, 67] Z: [19, 62, 67, 110] X: [63, 65, 66, 68] Z: [0, 20, 63, 68] X: [64, 66, 67, 69] Z: [1, 21, 64, 69] X: [65, 67, 68, 70] Z: [2, 22, 65, 70] X: [66, 68, 69, 71] Z: [3, 23, 66, 71] X: [67, 69, 70, 72] Z: [4, 24, 67, 72] X: [68, 70, 71, 73] Z: [5, 25, 68, 73] X: [69, 71, 72, 74] Z: [6, 26, 69, 74] X: [70, 72, 73, 75] Z: [7, 27, 70, 75] X: [71, 73, 74, 76] Z: [8, 28, 71, 76] X: [72, 74, 75, 77] Z: [9, 29, 72, 77] X: [73, 75, 76, 78] Z: [10, 30, 73, 78] X: [74, 76, 77, 79] Z: [11, 31, 74, 79] X: [75, 77, 78, 80] Z: [12, 32, 75, 80] X: [76, 78, 79, 81] Z: [13, 33, 76, 81] X: [77, 79, 80, 82] Z: [14, 34, 77, 82] X: [78, 80, 81, 83] Z: [15, 35, 78, 83] X: [79, 81, 82, 84] Z: [16, 36, 79, 84] X: [80, 82, 83, 85] Z: [17, 37, 80, 85] X: [81, 83, 84, 86] Z: [18, 38, 81, 86] X: [82, 84, 85, 87] Z: [19, 39, 82, 87] X: [83, 85, 86, 88] Z: [20, 40, 83, 88] X: [84, 86, 87, 89] Z: [21, 41, 84, 89] X: [85, 87, 88, 90] Z: [22, 42, 85, 90] X: [86, 88, 89, 91] Z: [23, 43, 86, 91] X: [87, 89, 90, 92] Z: [24, 44, 87, 92] X: [88, 90, 91, 93] Z: [25, 45, 88, 93] X: [89, 91, 92, 94] Z: [26, 46, 89, 94] X: [90, 92, 93, 95] Z: [27, 47, 90, 95] X: [91, 93, 94, 96] Z: [28, 48, 91, 96] X: [92, 94, 95, 97] Z: [29, 49, 92, 97] X: [93, 95, 96, 98] Z: [30, 50, 93, 98] X: [94, 96, 97, 99] Z: [31, 51, 94, 99] X: [95, 97, 98, 100] Z: [32, 52, 95, 100] X: [96, 98, 99, 101] Z: [33, 53, 96, 101] X: [97, 99, 100, 102] Z: [34, 54, 97, 102] X: [98, 100, 101, 103] Z: [35, 55, 98, 103] X: [99, 101, 102, 104] Z: [36, 56, 99, 104] X: [100, 102, 103, 105] Z: [37, 57, 100, 105] X: [101, 103, 104, 106] Z: [38, 58, 101, 106] X: [102, 104, 105, 107] Z: [39, 59, 102, 107] X: [103, 105, 106, 108] Z: [40, 60, 103, 108] X: [104, 106, 107, 109] Z: [41, 61, 104, 109] X: [105, 107, 108, 110] Z: [42, 62, 105, 110] X: [0, 106, 108, 109] Z: [0, 43, 63, 106] X: [1, 107, 109, 110] Z: [1, 44, 64, 107] X: [0, 2, 108, 110] Z: [2, 45, 65, 108] X: [0, 1, 3, 109] Z: [3, 46, 66, 109] X: [1, 2, 4, 110] Z: [4, 47, 67, 110] X: [0, 2, 3, 5] Z: [0, 5, 48, 68] X: [1, 3, 4, 6] Z: [1, 6, 49, 69] X: [2, 4, 5, 7] Z: [2, 7, 50, 70] X: [3, 5, 6, 8] Z: [3, 8, 51, 71] X: [4, 6, 7, 9] Z: [4, 9, 52, 72] X: [5, 7, 8, 10] Z: [5, 10, 53, 73] X: [6, 8, 9, 11] Z: [6, 11, 54, 74] X: [7, 9, 10, 12] Z: [7, 12, 55, 75] X: [8, 10, 11, 13] Z: [8, 13, 56, 76] X: [9, 11, 12, 14] Z: [9, 14, 57, 77] X: [10, 12, 13, 15] Z: [10, 15, 58, 78] X: [11, 13, 14, 16] Z: [11, 16, 59, 79] X: [12, 14, 15, 17] Z: [12, 17, 60, 80] X: [13, 15, 16, 18] Z: [13, 18, 61, 81] X: [14, 16, 17, 19] Z: [14, 19, 62, 82] X: [15, 17, 18, 20] Z: [15, 20, 63, 83] X: [16, 18, 19, 21] Z: [16, 21, 64, 84] X: [17, 19, 20, 22] Z: [17, 22, 65, 85] X: [18, 20, 21, 23] Z: [18, 23, 66, 86] X: [19, 21, 22, 24] Z: [19, 24, 67, 87] X: [20, 22, 23, 25] Z: [20, 25, 68, 88] X: [21, 23, 24, 26] Z: [21, 26, 69, 89] X: [22, 24, 25, 27] Z: [22, 27, 70, 90] X: [23, 25, 26, 28] Z: [23, 28, 71, 91] X: [24, 26, 27, 29] Z: [24, 29, 72, 92] X: [25, 27, 28, 30] Z: [25, 30, 73, 93] X: [26, 28, 29, 31] Z: [26, 31, 74, 94] X: [27, 29, 30, 32] Z: [27, 32, 75, 95] X: [28, 30, 31, 33] Z: [28, 33, 76, 96] X: [29, 31, 32, 34] Z: [29, 34, 77, 97] X: [30, 32, 33, 35] Z: [30, 35, 78, 98] X: [31, 33, 34, 36] Z: [31, 36, 79, 99] X: [32, 34, 35, 37] Z: [32, 37, 80, 100] X: [33, 35, 36, 38] Z: [33, 38, 81, 101] X: [34, 36, 37, 39] Z: [34, 39, 82, 102] X: [35, 37, 38, 40] Z: [35, 40, 83, 103] X: [36, 38, 39, 41] Z: [36, 41, 84, 104] X: [37, 39, 40, 42] Z: [37, 42, 85, 105] X: [38, 40, 41, 43] Z: [38, 43, 86, 106] X: [39, 41, 42, 44] Z: [39, 44, 87, 107] X: [40, 42, 43, 45] Z: [40, 45, 88, 108] X: [41, 43, 44, 46] Z: [41, 46, 89, 109] X: [42, 44, 45, 47] Z: [42, 47, 90, 110] X: [43, 45, 46, 48] Z: [0, 43, 48, 91] X: [44, 46, 47, 49] Z: [1, 44, 49, 92] X: [45, 47, 48, 50] Z: [2, 45, 50, 93] X: [46, 48, 49, 51] Z: [3, 46, 51, 94] X: [47, 49, 50, 52] Z: [4, 47, 52, 95] X: [48, 50, 51, 53] Z: [5, 48, 53, 96] X: [49, 51, 52, 54] Z: [6, 49, 54, 97] X: [50, 52, 53, 55] Z: [7, 50, 55, 98] X: [51, 53, 54, 56] Z: [8, 51, 56, 99] X: [52, 54, 55, 57] Z: [9, 52, 57, 100]

Code ID 111-3-12 · download JSON · raw on GitHub