[[151,1,21]] · QEC Challenge

[[151,1,21]] d ≤stabilizer

151

kd²/n

2.921

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 21 · witness Pauli weight 21 (5 Y factors; Hamming weight over 2n bits 26) (claimed upper_bound)

witness operator (Pauli string, 21 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)×151 (2,8)×906 (3,10)×5738 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 151 (2,8): 906 (2,10): 453 (3,10): 5738 (3,12): 9060 (3,14): 2265

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 151 cyclic shifts, a(x) = x⁷² + x⁷⁵ + x⁷⁶ + x⁷⁹, b(x) = x⁵³ + x⁷⁵ + x⁷⁶ + x⁹⁸ in F_2[x]/(x¹⁵¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁵¹ - 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_151, 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 29->21). 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

[[151,1,21]] — 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 <= 21 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 29->21).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 21.

Claim: d <= 21, 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 = 151; a(x) = x^72 + x^75 + x^76 + x^79; b(x) = x^53 + x^75 + x^76 + 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) = 1. Witness: X on [5, 11, 15, 28, 51, 59, 63, 77, 81, 86, 109, 122, 126, 132, 144], Z on [4, 11, 37, 55, 59, 77, 126, 133, 140, 144, 148].

Stabilizer generators

generators 151 (max weight 6; 151 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 (151, Pauli strings on 151 qubits)

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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIXIIYYIIXIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

symplectic rows (A | B) (151, sparse supports)

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

Code ID 151-1-21 · download JSON · raw on GitHub