[[121,11,8]] · QEC Challenge

[[121,11,8]] d ≤stabilizer

121

kd²/n

5.818

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 8 · witness Pauli weight 8 (2 Y factors; Hamming weight over 2n bits 10) (claimed upper_bound)

witness operator (Pauli string, 8 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)×121 (2,8)×121 (3,12)×2420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 121 (2,8): 121 (2,10): 121 (2,12): 1089 (2,14): 363 (3,12): 2420 (3,14): 4598 (3,16): 13673 (3,18): 10890 (3,20): 1815

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 121 cyclic shifts, a(x) = x⁵⁴ + x⁵⁶ + x⁶⁵ + x⁶⁷, b(x) = x⁴⁰ + x⁴⁸ + x⁷³ + x⁸¹ in F_2[x]/(x¹²¹ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x¹²¹ - 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_121, 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

[[121,11,8]] — 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 <= 8 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 8.

Claim: d <= 8, 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 = 121; a(x) = x^54 + x^56 + x^65 + x^67; b(x) = x^40 + x^48 + x^73 + x^81. 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) = 11. Witness: X on [51, 53, 59, 61], Z on [37, 48, 53, 59, 64, 75].

Stabilizer generators

generators 121 (max weight 8; 121 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 (121, Pauli strings on 121 qubits)

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symplectic rows (A | B) (121, sparse supports)

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

Code ID 121-11-8 · download JSON · raw on GitHub