[[161,23,6]] · QEC Challenge

[[161,23,6]] d ≤stabilizer

161

kd²/n

5.143

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 6

qubit degrees S 6

trapping sets S (1,6)×161 (2,8)×966 (3,6)×483 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 161 (2,8): 966 (2,10): 483 (3,6): 483 (3,8): 966 (3,10): 4669 (3,12): 5796 (3,14): 1449

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 161 cyclic shifts, a(x) = x⁶⁸ + x⁷⁰ + x⁹¹ + x⁹³, b(x) = x²⁴ + x⁷⁰ + x⁹¹ + x¹³⁷ in F_2[x]/(x¹⁶¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁶¹ - 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_161, 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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[161,23,6]] — 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 <= 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 = 161; a(x) = x^68 + x^70 + x^91 + x^93; b(x) = x^24 + x^70 + x^91 + x^137. 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) = 23. Witness: X on [67, 69], Z on [23, 46, 90, 113].

Stabilizer generators

generators 161 (max weight 6; 161 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 (161, Pauli strings on 161 qubits)

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IIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIYIIIIIIIIIIIIIIIIIIIIYIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIII

symplectic rows (A | B) (161, sparse supports)

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

Code ID 161-23-6 · download JSON · raw on GitHub