[[173,1,23]] · QEC Challenge

[[173,1,23]] d ≤stabilizer

173

kd²/n

3.058

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 23 · witness Pauli weight 23 (6 Y factors; Hamming weight over 2n bits 29) (claimed upper_bound)

witness operator (Pauli string, 23 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)×173 (2,8)×1038 (3,10)×6574 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 173 (2,8): 1038 (2,10): 519 (3,10): 6574 (3,12): 10380 (3,14): 2595

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 173 cyclic shifts, a(x) = x⁸³ + x⁸⁴ + x⁸⁹ + x⁹⁰, b(x) = x¹¹ + x⁸⁴ + x⁸⁹ + x¹⁶² in F_2[x]/(x¹⁷³ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁷³ - 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_173, 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 33->23). 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

[[173,1,23]] — 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 <= 23 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 33->23).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 23.

Claim: d <= 23, 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 = 173; a(x) = x^83 + x^84 + x^89 + x^90; b(x) = x^11 + x^84 + x^89 + x^162. 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) = 1. Witness: X on [31, 33, 54, 56, 71, 73, 88, 89, 104, 105, 132, 134, 149, 151, 166, 168], Z on [1, 7, 10, 13, 19, 25, 31, 33, 89, 105, 132, 133, 168].

Stabilizer generators

generators 173 (max weight 6; 173 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 (173, Pauli strings on 173 qubits)

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symplectic rows (A | B) (173, sparse supports)

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

Code ID 173-1-23 · download JSON · raw on GitHub