[[143,23,7]] · QEC Challenge

[[143,23,7]] d ≤stabilizer

143

kd²/n

7.881

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 7 · witness Pauli weight 7 (1 Y factor; Hamming weight over 2n bits 8) (claimed upper_bound)

witness operator (Pauli string, 7 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 8

qubit degrees S 8

trapping sets S (1,8)×143 (2,12)×1716 (3,14)×572 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 143 (2,12): 1716 (2,14): 572 (3,14): 572 (3,16): 26598 (3,18): 22308 (3,20): 3718

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 143 cyclic shifts, a(x) = x⁵⁴ + x⁶⁷ + x⁷⁶ + x⁸⁹, b(x) = x²⁹ + x⁶² + x⁸¹ + x¹¹⁴ in F_2[x]/(x¹⁴³ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹⁴³ - 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_143, 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 ≤ 8 (computed)

How this code was found

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

[[143,23,7]] — one-block palindromic cyclic stabilizer code X^a Z^b, weight 8

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 <= 7 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 7.

Claim: d <= 7, 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 = 143; a(x) = x^54 + x^67 + x^76 + x^89; b(x) = x^29 + x^62 + x^81 + x^114. 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 [30, 82, 134], Z on [38, 60, 82, 104, 126].

Stabilizer generators

generators 143 (max weight 8; 143 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 (143, Pauli strings on 143 qubits)

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symplectic rows (A | B) (143, sparse supports)

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

Code ID 143-23-7 · download JSON · raw on GitHub