[[75,10,5]] · QEC Challenge

[[75,10,5]] d ≤stabilizer

kd²/n

3.333

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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 5 · witness Pauli weight 5 (claimed upper_bound)

witness operator (Pauli string, 5 qubits)

IIIIIIIIIIIIIXIIIIIIIIIIIIIIXIIIIIIIIIIIIIIXIIIIIIIIIIIIIIXIIIIIIIIIIIIIIXI X: [13, 28, 43, 58, 73] Z: []

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 7

qubit degrees S 7

trapping sets S (1,7)×75 (2,10)×675 (3,9)×75 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,7): 75 (2,10): 675 (2,12): 225 (3,9): 75 (3,11): 150 (3,13): 7125 (3,15): 6375 (3,17): 975

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) = 1 + x³⁵ + x⁴⁰, b(x) = x²⁹ + x³¹ + x⁴⁴ + x⁴⁶ 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 2026-10-01

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

[[75,10,5]] — one-block palindromic cyclic stabilizer code X^a Z^b, weight 7

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 <= 5 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 5.

Claim: d <= 5, 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) = 1 + x^35 + x^40; b(x) = x^29 + x^31 + x^44 + x^46. 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) = 10. Witness: X on [13, 28, 43, 58, 73], Z on [].

Stabilizer generators

generators 75 (max weight 7; 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: [0, 35, 40] Z: [29, 31, 44, 46] X: [1, 36, 41] Z: [30, 32, 45, 47] X: [2, 37, 42] Z: [31, 33, 46, 48] X: [3, 38, 43] Z: [32, 34, 47, 49] X: [4, 39, 44] Z: [33, 35, 48, 50] X: [5, 40, 45] Z: [34, 36, 49, 51] X: [6, 41, 46] Z: [35, 37, 50, 52] X: [7, 42, 47] Z: [36, 38, 51, 53] X: [8, 43, 48] Z: [37, 39, 52, 54] X: [9, 44, 49] Z: [38, 40, 53, 55] X: [10, 45, 50] Z: [39, 41, 54, 56] X: [11, 46, 51] Z: [40, 42, 55, 57] X: [12, 47, 52] Z: [41, 43, 56, 58] X: [13, 48, 53] Z: [42, 44, 57, 59] X: [14, 49, 54] Z: [43, 45, 58, 60] X: [15, 50, 55] Z: [44, 46, 59, 61] X: [16, 51, 56] Z: [45, 47, 60, 62] X: [17, 52, 57] Z: [46, 48, 61, 63] X: [18, 53, 58] Z: [47, 49, 62, 64] X: [19, 54, 59] Z: [48, 50, 63, 65] X: [20, 55, 60] Z: [49, 51, 64, 66] X: [21, 56, 61] Z: [50, 52, 65, 67] X: [22, 57, 62] Z: [51, 53, 66, 68] X: [23, 58, 63] Z: [52, 54, 67, 69] X: [24, 59, 64] Z: [53, 55, 68, 70] X: [25, 60, 65] Z: [54, 56, 69, 71] X: [26, 61, 66] Z: [55, 57, 70, 72] X: [27, 62, 67] Z: [56, 58, 71, 73] X: [28, 63, 68] Z: [57, 59, 72, 74] X: [29, 64, 69] Z: [0, 58, 60, 73] X: [30, 65, 70] Z: [1, 59, 61, 74] X: [31, 66, 71] Z: [0, 2, 60, 62] X: [32, 67, 72] Z: [1, 3, 61, 63] X: [33, 68, 73] Z: [2, 4, 62, 64] X: [34, 69, 74] Z: [3, 5, 63, 65] X: [0, 35, 70] Z: [4, 6, 64, 66] X: [1, 36, 71] Z: [5, 7, 65, 67] X: [2, 37, 72] Z: [6, 8, 66, 68] X: [3, 38, 73] Z: [7, 9, 67, 69] X: [4, 39, 74] Z: [8, 10, 68, 70] X: [0, 5, 40] Z: [9, 11, 69, 71] X: [1, 6, 41] Z: [10, 12, 70, 72] X: [2, 7, 42] Z: [11, 13, 71, 73] X: [3, 8, 43] Z: [12, 14, 72, 74] X: [4, 9, 44] Z: [0, 13, 15, 73] X: [5, 10, 45] Z: [1, 14, 16, 74] X: [6, 11, 46] Z: [0, 2, 15, 17] X: [7, 12, 47] Z: [1, 3, 16, 18] X: [8, 13, 48] Z: [2, 4, 17, 19] X: [9, 14, 49] Z: [3, 5, 18, 20] X: [10, 15, 50] Z: [4, 6, 19, 21] X: [11, 16, 51] Z: [5, 7, 20, 22] X: [12, 17, 52] Z: [6, 8, 21, 23] X: [13, 18, 53] Z: [7, 9, 22, 24] X: [14, 19, 54] Z: [8, 10, 23, 25] X: [15, 20, 55] Z: [9, 11, 24, 26] X: [16, 21, 56] Z: [10, 12, 25, 27] X: [17, 22, 57] Z: [11, 13, 26, 28] X: [18, 23, 58] Z: [12, 14, 27, 29] X: [19, 24, 59] Z: [13, 15, 28, 30] X: [20, 25, 60] Z: [14, 16, 29, 31] X: [21, 26, 61] Z: [15, 17, 30, 32] X: [22, 27, 62] Z: [16, 18, 31, 33] X: [23, 28, 63] Z: [17, 19, 32, 34] X: [24, 29, 64] Z: [18, 20, 33, 35] X: [25, 30, 65] Z: [19, 21, 34, 36] X: [26, 31, 66] Z: [20, 22, 35, 37] X: [27, 32, 67] Z: [21, 23, 36, 38] X: [28, 33, 68] Z: [22, 24, 37, 39] X: [29, 34, 69] Z: [23, 25, 38, 40] X: [30, 35, 70] Z: [24, 26, 39, 41] X: [31, 36, 71] Z: [25, 27, 40, 42] X: [32, 37, 72] Z: [26, 28, 41, 43] X: [33, 38, 73] Z: [27, 29, 42, 44] X: [34, 39, 74] Z: [28, 30, 43, 45]

Code ID 75-10-5 · download JSON · raw on GitHub