[[178,2,20]] · QEC Challenge

[[178,2,20]] d ≤stabilizer

178

kd²/n

4.494

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 20 · witness Pauli weight 20 (4 Y factors; Hamming weight over 2n bits 24) (claimed upper_bound)

witness operator (Pauli string, 20 qubits)

IIIXIIIIIIIIIIIIIIIIIIIIIIIYIZIZIZIYIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIYIZIZIYIZIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZZZZZZIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII X: [3, 27, 35, 76, 82, 108] Z: [27, 29, 31, 33, 35, 74, 76, 78, 80, 82, 84, 141, 142, 143, 144, 145, 146, 148]

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)×178 (2,8)×1068 (3,10)×6764 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 178 (2,8): 1068 (2,10): 534 (3,10): 6764 (3,12): 10680 (3,14): 2670

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 178 cyclic shifts, a(x) = x⁸⁸ + x⁹⁰, b(x) = x²⁵ + x⁸⁰ + x⁹⁸ + x¹⁵³ in F_2[x]/(x¹⁷⁸ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁷⁸ - 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_178, 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 24->20). 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

[[178,2,20]] — 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 <= 20 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 24->20).

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

Claim: d <= 20, 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 = 178; a(x) = x^88 + x^90; b(x) = x^25 + x^80 + x^98 + x^153. 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) = 2. Witness: X on [3, 27, 35, 76, 82, 108], Z on [27, 29, 31, 33, 35, 74, 76, 78, 80, 82, 84, 141, 142, 143, 144, 145, 146, 148].

Stabilizer generators

generators 178 (max weight 6; 178 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 (178, Pauli strings on 178 qubits)

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symplectic rows (A | B) (178, sparse supports)

X: [88, 90] Z: [25, 80, 98, 153] X: [89, 91] Z: [26, 81, 99, 154] X: [90, 92] Z: [27, 82, 100, 155] X: [91, 93] Z: [28, 83, 101, 156] X: [92, 94] Z: [29, 84, 102, 157] X: [93, 95] Z: [30, 85, 103, 158] X: [94, 96] Z: [31, 86, 104, 159] X: [95, 97] Z: [32, 87, 105, 160] X: [96, 98] Z: [33, 88, 106, 161] X: [97, 99] Z: [34, 89, 107, 162] X: [98, 100] Z: [35, 90, 108, 163] X: [99, 101] Z: [36, 91, 109, 164] X: [100, 102] Z: [37, 92, 110, 165] X: [101, 103] Z: [38, 93, 111, 166] X: [102, 104] Z: [39, 94, 112, 167] X: [103, 105] Z: [40, 95, 113, 168] X: [104, 106] Z: [41, 96, 114, 169] X: [105, 107] Z: [42, 97, 115, 170] X: [106, 108] Z: [43, 98, 116, 171] X: [107, 109] Z: [44, 99, 117, 172] X: [108, 110] Z: [45, 100, 118, 173] X: [109, 111] Z: [46, 101, 119, 174] X: [110, 112] Z: [47, 102, 120, 175] X: [111, 113] Z: [48, 103, 121, 176] X: [112, 114] Z: [49, 104, 122, 177] X: [113, 115] Z: [0, 50, 105, 123] X: [114, 116] Z: [1, 51, 106, 124] X: [115, 117] Z: [2, 52, 107, 125] X: [116, 118] Z: [3, 53, 108, 126] X: [117, 119] Z: [4, 54, 109, 127] X: [118, 120] Z: [5, 55, 110, 128] X: [119, 121] Z: [6, 56, 111, 129] X: [120, 122] Z: [7, 57, 112, 130] X: [121, 123] Z: [8, 58, 113, 131] X: [122, 124] Z: [9, 59, 114, 132] X: [123, 125] Z: [10, 60, 115, 133] X: [124, 126] Z: [11, 61, 116, 134] X: [125, 127] Z: [12, 62, 117, 135] X: [126, 128] Z: [13, 63, 118, 136] X: [127, 129] Z: [14, 64, 119, 137] X: [128, 130] Z: [15, 65, 120, 138] X: [129, 131] Z: [16, 66, 121, 139] X: [130, 132] Z: [17, 67, 122, 140] X: [131, 133] Z: [18, 68, 123, 141] X: [132, 134] Z: [19, 69, 124, 142] X: [133, 135] Z: [20, 70, 125, 143] X: [134, 136] Z: [21, 71, 126, 144] X: [135, 137] Z: [22, 72, 127, 145] X: [136, 138] Z: [23, 73, 128, 146] X: [137, 139] Z: [24, 74, 129, 147] X: [138, 140] Z: [25, 75, 130, 148] X: [139, 141] Z: [26, 76, 131, 149] X: [140, 142] Z: [27, 77, 132, 150] X: [141, 143] Z: [28, 78, 133, 151] X: [142, 144] Z: [29, 79, 134, 152] X: [143, 145] Z: [30, 80, 135, 153] X: [144, 146] Z: [31, 81, 136, 154] X: [145, 147] Z: [32, 82, 137, 155] X: [146, 148] Z: [33, 83, 138, 156] X: [147, 149] Z: [34, 84, 139, 157] X: [148, 150] Z: [35, 85, 140, 158] X: [149, 151] Z: [36, 86, 141, 159] X: [150, 152] Z: [37, 87, 142, 160] X: [151, 153] Z: [38, 88, 143, 161] X: [152, 154] Z: [39, 89, 144, 162] X: [153, 155] Z: [40, 90, 145, 163] X: [154, 156] Z: [41, 91, 146, 164] X: [155, 157] Z: [42, 92, 147, 165] X: [156, 158] Z: [43, 93, 148, 166] X: [157, 159] Z: [44, 94, 149, 167] X: [158, 160] Z: [45, 95, 150, 168] X: [159, 161] Z: [46, 96, 151, 169] X: [160, 162] Z: [47, 97, 152, 170] X: [161, 163] Z: [48, 98, 153, 171] X: [162, 164] Z: [49, 99, 154, 172] X: [163, 165] Z: [50, 100, 155, 173] X: [164, 166] Z: [51, 101, 156, 174] X: [165, 167] Z: [52, 102, 157, 175] X: [166, 168] Z: [53, 103, 158, 176] X: [167, 169] Z: [54, 104, 159, 177] X: [168, 170] Z: [0, 55, 105, 160] X: [169, 171] Z: [1, 56, 106, 161] X: [170, 172] Z: [2, 57, 107, 162] X: [171, 173] Z: [3, 58, 108, 163] X: [172, 174] Z: [4, 59, 109, 164] X: [173, 175] Z: [5, 60, 110, 165] X: [174, 176] Z: [6, 61, 111, 166] X: [175, 177] Z: [7, 62, 112, 167] X: [0, 176] Z: [8, 63, 113, 168] X: [1, 177] Z: [9, 64, 114, 169] X: [0, 2] Z: [10, 65, 115, 170] X: [1, 3] Z: [11, 66, 116, 171] X: [2, 4] Z: [12, 67, 117, 172] X: [3, 5] Z: [13, 68, 118, 173] X: [4, 6] Z: [14, 69, 119, 174] X: [5, 7] Z: [15, 70, 120, 175] X: [6, 8] Z: [16, 71, 121, 176] X: [7, 9] Z: [17, 72, 122, 177] X: [8, 10] Z: [0, 18, 73, 123] X: [9, 11] Z: [1, 19, 74, 124] X: [10, 12] Z: [2, 20, 75, 125] X: [11, 13] Z: [3, 21, 76, 126] X: [12, 14] Z: [4, 22, 77, 127] X: [13, 15] Z: [5, 23, 78, 128] X: [14, 16] Z: [6, 24, 79, 129] X: [15, 17] Z: [7, 25, 80, 130] X: [16, 18] Z: [8, 26, 81, 131] X: [17, 19] Z: [9, 27, 82, 132] X: [18, 20] Z: [10, 28, 83, 133] X: [19, 21] Z: [11, 29, 84, 134] X: [20, 22] Z: [12, 30, 85, 135] X: [21, 23] Z: [13, 31, 86, 136] X: [22, 24] Z: [14, 32, 87, 137] X: [23, 25] Z: [15, 33, 88, 138] X: [24, 26] Z: [16, 34, 89, 139] X: [25, 27] Z: [17, 35, 90, 140] X: [26, 28] Z: [18, 36, 91, 141] X: [27, 29] Z: [19, 37, 92, 142] X: [28, 30] Z: [20, 38, 93, 143] X: [29, 31] Z: [21, 39, 94, 144] X: [30, 32] Z: [22, 40, 95, 145] X: [31, 33] Z: [23, 41, 96, 146] X: [32, 34] Z: [24, 42, 97, 147] X: [33, 35] Z: [25, 43, 98, 148] X: [34, 36] Z: [26, 44, 99, 149] X: [35, 37] Z: [27, 45, 100, 150] X: [36, 38] Z: [28, 46, 101, 151] X: [37, 39] Z: [29, 47, 102, 152] X: [38, 40] Z: [30, 48, 103, 153] X: [39, 41] Z: [31, 49, 104, 154] X: [40, 42] Z: [32, 50, 105, 155] X: [41, 43] Z: [33, 51, 106, 156] X: [42, 44] Z: [34, 52, 107, 157] X: [43, 45] Z: [35, 53, 108, 158] X: [44, 46] Z: [36, 54, 109, 159] X: [45, 47] Z: [37, 55, 110, 160] X: [46, 48] Z: [38, 56, 111, 161] X: [47, 49] Z: [39, 57, 112, 162] X: [48, 50] Z: [40, 58, 113, 163] X: [49, 51] Z: [41, 59, 114, 164] X: [50, 52] Z: [42, 60, 115, 165] X: [51, 53] Z: [43, 61, 116, 166] X: [52, 54] Z: [44, 62, 117, 167] X: [53, 55] Z: [45, 63, 118, 168] X: [54, 56] Z: [46, 64, 119, 169] X: [55, 57] Z: [47, 65, 120, 170] X: [56, 58] Z: [48, 66, 121, 171] X: [57, 59] Z: [49, 67, 122, 172] X: [58, 60] Z: [50, 68, 123, 173] X: [59, 61] Z: [51, 69, 124, 174] X: [60, 62] Z: [52, 70, 125, 175] X: [61, 63] Z: [53, 71, 126, 176] X: [62, 64] Z: [54, 72, 127, 177] X: [63, 65] Z: [0, 55, 73, 128] X: [64, 66] Z: [1, 56, 74, 129] X: [65, 67] Z: [2, 57, 75, 130] X: [66, 68] Z: [3, 58, 76, 131] X: [67, 69] Z: [4, 59, 77, 132] X: [68, 70] Z: [5, 60, 78, 133] X: [69, 71] Z: [6, 61, 79, 134] X: [70, 72] Z: [7, 62, 80, 135] X: [71, 73] Z: [8, 63, 81, 136] X: [72, 74] Z: [9, 64, 82, 137] X: [73, 75] Z: [10, 65, 83, 138] X: [74, 76] Z: [11, 66, 84, 139] X: [75, 77] Z: [12, 67, 85, 140] X: [76, 78] Z: [13, 68, 86, 141] X: [77, 79] Z: [14, 69, 87, 142] X: [78, 80] Z: [15, 70, 88, 143] X: [79, 81] Z: [16, 71, 89, 144] X: [80, 82] Z: [17, 72, 90, 145] X: [81, 83] Z: [18, 73, 91, 146] X: [82, 84] Z: [19, 74, 92, 147] X: [83, 85] Z: [20, 75, 93, 148] X: [84, 86] Z: [21, 76, 94, 149] X: [85, 87] Z: [22, 77, 95, 150] X: [86, 88] Z: [23, 78, 96, 151] X: [87, 89] Z: [24, 79, 97, 152]

Code ID 178-2-20 · download JSON · raw on GitHub