[[113,1,17]] · QEC Challenge

[[113,1,17]] d ≤stabilizer

113

kd²/n

2.558

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 17 · witness Pauli weight 17 (7 Y factors; Hamming weight over 2n bits 24) (claimed upper_bound)

witness operator (Pauli string, 17 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)×113 (2,8)×678 (3,10)×4294 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 113 (2,8): 678 (2,10): 339 (3,10): 4294 (3,12): 6780 (3,14): 1695

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 113 cyclic shifts, a(x) = x⁵⁴ + x⁵⁵ + x⁵⁸ + x⁵⁹, b(x) = x⁷ + x⁵⁵ + x⁵⁸ + x¹⁰⁶ in F_2[x]/(x¹¹³ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹¹³ - 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_113, 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 21->17). 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

[[113,1,17]] — 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 <= 17 at 300 trials. Deep: 2 x 20,000 trials found nothing lighter (dropped 21->17).

qldpc submit re-searched (20,000 trials) and the filed witness has Pauli weight 17.

Claim: d <= 17, 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 = 113; a(x) = x^54 + x^55 + x^58 + x^59; b(x) = x^7 + x^55 + x^58 + x^106. 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, 24, 43, 57, 58, 60, 71, 90, 109, 111, 112], Z on [2, 7, 8, 24, 43, 50, 54, 57, 58, 59, 60, 71, 90].

Stabilizer generators

generators 113 (max weight 6; 113 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 (113, Pauli strings on 113 qubits)

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IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIZI IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIZ ZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIII IZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIII IIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIII IIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIII IIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIII IIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIII IIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXYIIYXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIII

symplectic rows (A | B) (113, sparse supports)

X: [54, 55, 58, 59] Z: [7, 55, 58, 106] X: [55, 56, 59, 60] Z: [8, 56, 59, 107] X: [56, 57, 60, 61] Z: [9, 57, 60, 108] X: [57, 58, 61, 62] Z: [10, 58, 61, 109] X: [58, 59, 62, 63] Z: [11, 59, 62, 110] X: [59, 60, 63, 64] Z: [12, 60, 63, 111] X: [60, 61, 64, 65] Z: [13, 61, 64, 112] X: [61, 62, 65, 66] Z: [0, 14, 62, 65] X: [62, 63, 66, 67] Z: [1, 15, 63, 66] X: [63, 64, 67, 68] Z: [2, 16, 64, 67] X: [64, 65, 68, 69] Z: [3, 17, 65, 68] X: [65, 66, 69, 70] Z: [4, 18, 66, 69] X: [66, 67, 70, 71] Z: [5, 19, 67, 70] X: [67, 68, 71, 72] Z: [6, 20, 68, 71] X: [68, 69, 72, 73] Z: [7, 21, 69, 72] X: [69, 70, 73, 74] Z: [8, 22, 70, 73] X: [70, 71, 74, 75] Z: [9, 23, 71, 74] X: [71, 72, 75, 76] Z: [10, 24, 72, 75] X: [72, 73, 76, 77] Z: [11, 25, 73, 76] X: [73, 74, 77, 78] Z: [12, 26, 74, 77] X: [74, 75, 78, 79] Z: [13, 27, 75, 78] X: [75, 76, 79, 80] Z: [14, 28, 76, 79] X: [76, 77, 80, 81] Z: [15, 29, 77, 80] X: [77, 78, 81, 82] Z: [16, 30, 78, 81] X: [78, 79, 82, 83] Z: [17, 31, 79, 82] X: [79, 80, 83, 84] Z: [18, 32, 80, 83] X: [80, 81, 84, 85] Z: [19, 33, 81, 84] X: [81, 82, 85, 86] Z: [20, 34, 82, 85] X: [82, 83, 86, 87] Z: [21, 35, 83, 86] X: [83, 84, 87, 88] Z: [22, 36, 84, 87] X: [84, 85, 88, 89] Z: [23, 37, 85, 88] X: [85, 86, 89, 90] Z: [24, 38, 86, 89] X: [86, 87, 90, 91] Z: [25, 39, 87, 90] X: [87, 88, 91, 92] Z: [26, 40, 88, 91] X: [88, 89, 92, 93] Z: [27, 41, 89, 92] X: [89, 90, 93, 94] Z: [28, 42, 90, 93] X: [90, 91, 94, 95] Z: [29, 43, 91, 94] X: [91, 92, 95, 96] Z: [30, 44, 92, 95] X: [92, 93, 96, 97] Z: [31, 45, 93, 96] X: [93, 94, 97, 98] Z: [32, 46, 94, 97] X: [94, 95, 98, 99] Z: [33, 47, 95, 98] X: [95, 96, 99, 100] Z: [34, 48, 96, 99] X: [96, 97, 100, 101] Z: [35, 49, 97, 100] X: [97, 98, 101, 102] Z: [36, 50, 98, 101] X: [98, 99, 102, 103] Z: [37, 51, 99, 102] X: [99, 100, 103, 104] Z: [38, 52, 100, 103] X: [100, 101, 104, 105] Z: [39, 53, 101, 104] X: [101, 102, 105, 106] Z: [40, 54, 102, 105] X: [102, 103, 106, 107] Z: [41, 55, 103, 106] X: [103, 104, 107, 108] Z: [42, 56, 104, 107] X: [104, 105, 108, 109] Z: [43, 57, 105, 108] X: [105, 106, 109, 110] Z: [44, 58, 106, 109] X: [106, 107, 110, 111] Z: [45, 59, 107, 110] X: [107, 108, 111, 112] Z: [46, 60, 108, 111] X: [0, 108, 109, 112] Z: [47, 61, 109, 112] X: [0, 1, 109, 110] Z: [0, 48, 62, 110] X: [1, 2, 110, 111] Z: [1, 49, 63, 111] X: [2, 3, 111, 112] Z: [2, 50, 64, 112] X: [0, 3, 4, 112] Z: [0, 3, 51, 65] X: [0, 1, 4, 5] Z: [1, 4, 52, 66] X: [1, 2, 5, 6] Z: [2, 5, 53, 67] X: [2, 3, 6, 7] Z: [3, 6, 54, 68] X: [3, 4, 7, 8] Z: [4, 7, 55, 69] X: [4, 5, 8, 9] Z: [5, 8, 56, 70] X: [5, 6, 9, 10] Z: [6, 9, 57, 71] X: [6, 7, 10, 11] Z: [7, 10, 58, 72] X: [7, 8, 11, 12] Z: [8, 11, 59, 73] X: [8, 9, 12, 13] Z: [9, 12, 60, 74] X: [9, 10, 13, 14] Z: [10, 13, 61, 75] X: [10, 11, 14, 15] Z: [11, 14, 62, 76] X: [11, 12, 15, 16] Z: [12, 15, 63, 77] X: [12, 13, 16, 17] Z: [13, 16, 64, 78] X: [13, 14, 17, 18] Z: [14, 17, 65, 79] X: [14, 15, 18, 19] Z: [15, 18, 66, 80] X: [15, 16, 19, 20] Z: [16, 19, 67, 81] X: [16, 17, 20, 21] Z: [17, 20, 68, 82] X: [17, 18, 21, 22] Z: [18, 21, 69, 83] X: [18, 19, 22, 23] Z: [19, 22, 70, 84] X: [19, 20, 23, 24] Z: [20, 23, 71, 85] X: [20, 21, 24, 25] Z: [21, 24, 72, 86] X: [21, 22, 25, 26] Z: [22, 25, 73, 87] X: [22, 23, 26, 27] Z: [23, 26, 74, 88] X: [23, 24, 27, 28] Z: [24, 27, 75, 89] X: [24, 25, 28, 29] Z: [25, 28, 76, 90] X: [25, 26, 29, 30] Z: [26, 29, 77, 91] X: [26, 27, 30, 31] Z: [27, 30, 78, 92] X: [27, 28, 31, 32] Z: [28, 31, 79, 93] X: [28, 29, 32, 33] Z: [29, 32, 80, 94] X: [29, 30, 33, 34] Z: [30, 33, 81, 95] X: [30, 31, 34, 35] Z: [31, 34, 82, 96] X: [31, 32, 35, 36] Z: [32, 35, 83, 97] X: [32, 33, 36, 37] Z: [33, 36, 84, 98] X: [33, 34, 37, 38] Z: [34, 37, 85, 99] X: [34, 35, 38, 39] Z: [35, 38, 86, 100] X: [35, 36, 39, 40] Z: [36, 39, 87, 101] X: [36, 37, 40, 41] Z: [37, 40, 88, 102] X: [37, 38, 41, 42] Z: [38, 41, 89, 103] X: [38, 39, 42, 43] Z: [39, 42, 90, 104] X: [39, 40, 43, 44] Z: [40, 43, 91, 105] X: [40, 41, 44, 45] Z: [41, 44, 92, 106] X: [41, 42, 45, 46] Z: [42, 45, 93, 107] X: [42, 43, 46, 47] Z: [43, 46, 94, 108] X: [43, 44, 47, 48] Z: [44, 47, 95, 109] X: [44, 45, 48, 49] Z: [45, 48, 96, 110] X: [45, 46, 49, 50] Z: [46, 49, 97, 111] X: [46, 47, 50, 51] Z: [47, 50, 98, 112] X: [47, 48, 51, 52] Z: [0, 48, 51, 99] X: [48, 49, 52, 53] Z: [1, 49, 52, 100] X: [49, 50, 53, 54] Z: [2, 50, 53, 101] X: [50, 51, 54, 55] Z: [3, 51, 54, 102] X: [51, 52, 55, 56] Z: [4, 52, 55, 103] X: [52, 53, 56, 57] Z: [5, 53, 56, 104] X: [53, 54, 57, 58] Z: [6, 54, 57, 105]

Code ID 113-1-17 · download JSON · raw on GitHub