[[123,3,14]] · QEC Challenge

[[123,3,14]] d ≤stabilizer

123

kd²/n

4.78

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 14 · witness Pauli weight 14 (4 Y factors; Hamming weight over 2n bits 18) (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)×123 (2,8)×738 (3,10)×4674 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 123 (2,8): 738 (2,10): 369 (3,10): 4674 (3,12): 7380 (3,14): 1845

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 123 cyclic shifts, a(x) = x⁵⁵ + x⁵⁹ + x⁶⁴ + x⁶⁸, b(x) = x²⁵ + x⁵⁵ + x⁶⁸ + x⁹⁸ in F_2[x]/(x¹²³ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹²³ - 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_123, 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 15->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

[[123,3,14]] — 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 15->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 = 123; a(x) = x^55 + x^59 + x^64 + x^68; b(x) = x^25 + x^55 + x^68 + x^98. 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 [0, 53, 57, 119], Z on [0, 5, 14, 39, 48, 53, 57, 62, 71, 80, 96, 105, 114, 119].

Stabilizer generators

generators 123 (max weight 6; 123 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 (123, Pauli strings on 123 qubits)

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IIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIXIIIIXIIIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII 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symplectic rows (A | B) (123, sparse supports)

X: [55, 59, 64, 68] Z: [25, 55, 68, 98] X: [56, 60, 65, 69] Z: [26, 56, 69, 99] X: [57, 61, 66, 70] Z: [27, 57, 70, 100] X: [58, 62, 67, 71] Z: [28, 58, 71, 101] X: [59, 63, 68, 72] Z: [29, 59, 72, 102] X: [60, 64, 69, 73] Z: [30, 60, 73, 103] X: [61, 65, 70, 74] Z: [31, 61, 74, 104] X: [62, 66, 71, 75] Z: [32, 62, 75, 105] X: [63, 67, 72, 76] Z: [33, 63, 76, 106] X: [64, 68, 73, 77] Z: [34, 64, 77, 107] X: [65, 69, 74, 78] Z: [35, 65, 78, 108] X: [66, 70, 75, 79] Z: [36, 66, 79, 109] X: [67, 71, 76, 80] Z: [37, 67, 80, 110] X: [68, 72, 77, 81] Z: [38, 68, 81, 111] X: [69, 73, 78, 82] Z: [39, 69, 82, 112] X: [70, 74, 79, 83] Z: [40, 70, 83, 113] X: [71, 75, 80, 84] Z: [41, 71, 84, 114] X: [72, 76, 81, 85] Z: [42, 72, 85, 115] X: [73, 77, 82, 86] Z: [43, 73, 86, 116] X: [74, 78, 83, 87] Z: [44, 74, 87, 117] X: [75, 79, 84, 88] Z: [45, 75, 88, 118] X: [76, 80, 85, 89] Z: [46, 76, 89, 119] X: [77, 81, 86, 90] Z: [47, 77, 90, 120] X: [78, 82, 87, 91] Z: [48, 78, 91, 121] X: [79, 83, 88, 92] Z: [49, 79, 92, 122] X: [80, 84, 89, 93] Z: [0, 50, 80, 93] X: [81, 85, 90, 94] Z: [1, 51, 81, 94] X: [82, 86, 91, 95] Z: [2, 52, 82, 95] X: [83, 87, 92, 96] Z: [3, 53, 83, 96] X: [84, 88, 93, 97] Z: [4, 54, 84, 97] X: [85, 89, 94, 98] Z: [5, 55, 85, 98] X: [86, 90, 95, 99] Z: [6, 56, 86, 99] X: [87, 91, 96, 100] Z: [7, 57, 87, 100] X: [88, 92, 97, 101] Z: [8, 58, 88, 101] X: [89, 93, 98, 102] Z: [9, 59, 89, 102] X: [90, 94, 99, 103] Z: [10, 60, 90, 103] X: [91, 95, 100, 104] Z: [11, 61, 91, 104] X: [92, 96, 101, 105] Z: [12, 62, 92, 105] X: [93, 97, 102, 106] Z: [13, 63, 93, 106] X: [94, 98, 103, 107] Z: [14, 64, 94, 107] X: [95, 99, 104, 108] Z: [15, 65, 95, 108] X: [96, 100, 105, 109] Z: [16, 66, 96, 109] X: [97, 101, 106, 110] Z: [17, 67, 97, 110] X: [98, 102, 107, 111] Z: [18, 68, 98, 111] X: [99, 103, 108, 112] Z: [19, 69, 99, 112] X: [100, 104, 109, 113] Z: [20, 70, 100, 113] X: [101, 105, 110, 114] Z: [21, 71, 101, 114] X: [102, 106, 111, 115] Z: [22, 72, 102, 115] X: [103, 107, 112, 116] Z: [23, 73, 103, 116] X: [104, 108, 113, 117] Z: [24, 74, 104, 117] X: [105, 109, 114, 118] Z: [25, 75, 105, 118] X: [106, 110, 115, 119] Z: [26, 76, 106, 119] X: [107, 111, 116, 120] Z: [27, 77, 107, 120] X: [108, 112, 117, 121] Z: [28, 78, 108, 121] X: [109, 113, 118, 122] Z: [29, 79, 109, 122] X: [0, 110, 114, 119] Z: [0, 30, 80, 110] X: [1, 111, 115, 120] Z: [1, 31, 81, 111] X: [2, 112, 116, 121] Z: [2, 32, 82, 112] X: [3, 113, 117, 122] Z: [3, 33, 83, 113] X: [0, 4, 114, 118] Z: [4, 34, 84, 114] X: [1, 5, 115, 119] Z: [5, 35, 85, 115] X: [2, 6, 116, 120] Z: [6, 36, 86, 116] X: [3, 7, 117, 121] Z: [7, 37, 87, 117] X: [4, 8, 118, 122] Z: [8, 38, 88, 118] X: [0, 5, 9, 119] Z: [9, 39, 89, 119] X: [1, 6, 10, 120] Z: [10, 40, 90, 120] X: [2, 7, 11, 121] Z: [11, 41, 91, 121] X: [3, 8, 12, 122] Z: [12, 42, 92, 122] X: [0, 4, 9, 13] Z: [0, 13, 43, 93] X: [1, 5, 10, 14] Z: [1, 14, 44, 94] X: [2, 6, 11, 15] Z: [2, 15, 45, 95] X: [3, 7, 12, 16] Z: [3, 16, 46, 96] X: [4, 8, 13, 17] Z: [4, 17, 47, 97] X: [5, 9, 14, 18] Z: [5, 18, 48, 98] X: [6, 10, 15, 19] Z: [6, 19, 49, 99] X: [7, 11, 16, 20] Z: [7, 20, 50, 100] X: [8, 12, 17, 21] Z: [8, 21, 51, 101] X: [9, 13, 18, 22] Z: [9, 22, 52, 102] X: [10, 14, 19, 23] Z: [10, 23, 53, 103] X: [11, 15, 20, 24] Z: [11, 24, 54, 104] X: [12, 16, 21, 25] Z: [12, 25, 55, 105] X: [13, 17, 22, 26] Z: [13, 26, 56, 106] X: [14, 18, 23, 27] Z: [14, 27, 57, 107] X: [15, 19, 24, 28] Z: [15, 28, 58, 108] X: [16, 20, 25, 29] Z: [16, 29, 59, 109] X: [17, 21, 26, 30] Z: [17, 30, 60, 110] X: [18, 22, 27, 31] Z: [18, 31, 61, 111] X: [19, 23, 28, 32] Z: [19, 32, 62, 112] X: [20, 24, 29, 33] Z: [20, 33, 63, 113] X: [21, 25, 30, 34] Z: [21, 34, 64, 114] X: [22, 26, 31, 35] Z: [22, 35, 65, 115] X: [23, 27, 32, 36] Z: [23, 36, 66, 116] X: [24, 28, 33, 37] Z: [24, 37, 67, 117] X: [25, 29, 34, 38] Z: [25, 38, 68, 118] X: [26, 30, 35, 39] Z: [26, 39, 69, 119] X: [27, 31, 36, 40] Z: [27, 40, 70, 120] X: [28, 32, 37, 41] Z: [28, 41, 71, 121] X: [29, 33, 38, 42] Z: [29, 42, 72, 122] X: [30, 34, 39, 43] Z: [0, 30, 43, 73] X: [31, 35, 40, 44] Z: [1, 31, 44, 74] X: [32, 36, 41, 45] Z: [2, 32, 45, 75] X: [33, 37, 42, 46] Z: [3, 33, 46, 76] X: [34, 38, 43, 47] Z: [4, 34, 47, 77] X: [35, 39, 44, 48] Z: [5, 35, 48, 78] X: [36, 40, 45, 49] Z: [6, 36, 49, 79] X: [37, 41, 46, 50] Z: [7, 37, 50, 80] X: [38, 42, 47, 51] Z: [8, 38, 51, 81] X: [39, 43, 48, 52] Z: [9, 39, 52, 82] X: [40, 44, 49, 53] Z: [10, 40, 53, 83] X: [41, 45, 50, 54] Z: [11, 41, 54, 84] X: [42, 46, 51, 55] Z: [12, 42, 55, 85] X: [43, 47, 52, 56] Z: [13, 43, 56, 86] X: [44, 48, 53, 57] Z: [14, 44, 57, 87] X: [45, 49, 54, 58] Z: [15, 45, 58, 88] X: [46, 50, 55, 59] Z: [16, 46, 59, 89] X: [47, 51, 56, 60] Z: [17, 47, 60, 90] X: [48, 52, 57, 61] Z: [18, 48, 61, 91] X: [49, 53, 58, 62] Z: [19, 49, 62, 92] X: [50, 54, 59, 63] Z: [20, 50, 63, 93] X: [51, 55, 60, 64] Z: [21, 51, 64, 94] X: [52, 56, 61, 65] Z: [22, 52, 65, 95] X: [53, 57, 62, 66] Z: [23, 53, 66, 96] X: [54, 58, 63, 67] Z: [24, 54, 67, 97]

Code ID 123-3-14 · download JSON · raw on GitHub