[[75,7,10]] · QEC Challenge

[[75,7,10]] d ≤stabilizer

kd²/n

9.333

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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 10 · witness Pauli weight 10 (claimed upper_bound)

witness operator (Pauli string, 10 qubits)

IIIIIIXIIIIIXIIIIIIIZIIIIIXIIIIIXIIIIIIIIXIIIIIXIIIIIZIIIIIIIXIIIIIXIIIIIII X: [6, 12, 26, 32, 41, 47, 61, 67] Z: [20, 53]

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)×75 (2,8)×75 (3,12)×1800 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 75 (2,8): 75 (2,10): 75 (2,12): 675 (2,14): 225 (3,12): 1800 (3,14): 3075 (3,16): 7725 (3,18): 5475 (3,20): 675

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 75 cyclic shifts, a(x) = x³² + x³⁷ + x³⁸ + x⁴³, b(x) = x¹¹ + x³¹ + x⁴⁴ + x⁶⁴ in F_2[x]/(x⁷⁵ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁷⁵ - 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_75, 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

[[75,7,10]] — 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 <= 10 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 10.

Claim: d <= 10, 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 = 75; a(x) = x^32 + x^37 + x^38 + x^43; b(x) = x^11 + x^31 + x^44 + x^64. 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) = 7. Witness: X on [6, 12, 26, 32, 41, 47, 61, 67], Z on [20, 53].

Stabilizer generators

generators 75 (max weight 8; 75 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 (75, Pauli strings on 75 qubits)

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symplectic rows (A | B) (75, sparse supports)

X: [32, 37, 38, 43] Z: [11, 31, 44, 64] X: [33, 38, 39, 44] Z: [12, 32, 45, 65] X: [34, 39, 40, 45] Z: [13, 33, 46, 66] X: [35, 40, 41, 46] Z: [14, 34, 47, 67] X: [36, 41, 42, 47] Z: [15, 35, 48, 68] X: [37, 42, 43, 48] Z: [16, 36, 49, 69] X: [38, 43, 44, 49] Z: [17, 37, 50, 70] X: [39, 44, 45, 50] Z: [18, 38, 51, 71] X: [40, 45, 46, 51] Z: [19, 39, 52, 72] X: [41, 46, 47, 52] Z: [20, 40, 53, 73] X: [42, 47, 48, 53] Z: [21, 41, 54, 74] X: [43, 48, 49, 54] Z: [0, 22, 42, 55] X: [44, 49, 50, 55] Z: [1, 23, 43, 56] X: [45, 50, 51, 56] Z: [2, 24, 44, 57] X: [46, 51, 52, 57] Z: [3, 25, 45, 58] X: [47, 52, 53, 58] Z: [4, 26, 46, 59] X: [48, 53, 54, 59] Z: [5, 27, 47, 60] X: [49, 54, 55, 60] Z: [6, 28, 48, 61] X: [50, 55, 56, 61] Z: [7, 29, 49, 62] X: [51, 56, 57, 62] Z: [8, 30, 50, 63] X: [52, 57, 58, 63] Z: [9, 31, 51, 64] X: [53, 58, 59, 64] Z: [10, 32, 52, 65] X: [54, 59, 60, 65] Z: [11, 33, 53, 66] X: [55, 60, 61, 66] Z: [12, 34, 54, 67] X: [56, 61, 62, 67] Z: [13, 35, 55, 68] X: [57, 62, 63, 68] Z: [14, 36, 56, 69] X: [58, 63, 64, 69] Z: [15, 37, 57, 70] X: [59, 64, 65, 70] Z: [16, 38, 58, 71] X: [60, 65, 66, 71] Z: [17, 39, 59, 72] X: [61, 66, 67, 72] Z: [18, 40, 60, 73] X: [62, 67, 68, 73] Z: [19, 41, 61, 74] X: [63, 68, 69, 74] Z: [0, 20, 42, 62] X: [0, 64, 69, 70] Z: [1, 21, 43, 63] X: [1, 65, 70, 71] Z: [2, 22, 44, 64] X: [2, 66, 71, 72] Z: [3, 23, 45, 65] X: [3, 67, 72, 73] Z: [4, 24, 46, 66] X: [4, 68, 73, 74] Z: [5, 25, 47, 67] X: [0, 5, 69, 74] Z: [6, 26, 48, 68] X: [0, 1, 6, 70] Z: [7, 27, 49, 69] X: [1, 2, 7, 71] Z: [8, 28, 50, 70] X: [2, 3, 8, 72] Z: [9, 29, 51, 71] X: [3, 4, 9, 73] Z: [10, 30, 52, 72] X: [4, 5, 10, 74] Z: [11, 31, 53, 73] X: [0, 5, 6, 11] Z: [12, 32, 54, 74] X: [1, 6, 7, 12] Z: [0, 13, 33, 55] X: [2, 7, 8, 13] Z: [1, 14, 34, 56] X: [3, 8, 9, 14] Z: [2, 15, 35, 57] X: [4, 9, 10, 15] Z: [3, 16, 36, 58] X: [5, 10, 11, 16] Z: [4, 17, 37, 59] X: [6, 11, 12, 17] Z: [5, 18, 38, 60] X: [7, 12, 13, 18] Z: [6, 19, 39, 61] X: [8, 13, 14, 19] Z: [7, 20, 40, 62] X: [9, 14, 15, 20] Z: [8, 21, 41, 63] X: [10, 15, 16, 21] Z: [9, 22, 42, 64] X: [11, 16, 17, 22] Z: [10, 23, 43, 65] X: [12, 17, 18, 23] Z: [11, 24, 44, 66] X: [13, 18, 19, 24] Z: [12, 25, 45, 67] X: [14, 19, 20, 25] Z: [13, 26, 46, 68] X: [15, 20, 21, 26] Z: [14, 27, 47, 69] X: [16, 21, 22, 27] Z: [15, 28, 48, 70] X: [17, 22, 23, 28] Z: [16, 29, 49, 71] X: [18, 23, 24, 29] Z: [17, 30, 50, 72] X: [19, 24, 25, 30] Z: [18, 31, 51, 73] X: [20, 25, 26, 31] Z: [19, 32, 52, 74] X: [21, 26, 27, 32] Z: [0, 20, 33, 53] X: [22, 27, 28, 33] Z: [1, 21, 34, 54] X: [23, 28, 29, 34] Z: [2, 22, 35, 55] X: [24, 29, 30, 35] Z: [3, 23, 36, 56] X: [25, 30, 31, 36] Z: [4, 24, 37, 57] X: [26, 31, 32, 37] Z: [5, 25, 38, 58] X: [27, 32, 33, 38] Z: [6, 26, 39, 59] X: [28, 33, 34, 39] Z: [7, 27, 40, 60] X: [29, 34, 35, 40] Z: [8, 28, 41, 61] X: [30, 35, 36, 41] Z: [9, 29, 42, 62] X: [31, 36, 37, 42] Z: [10, 30, 43, 63]

Code ID 75-7-10 · download JSON · raw on GitHub