[[77,11,7]] · QEC Challenge

[[77,11,7]] d ≤stabilizer

kd²/n

7.0

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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 7 · witness Pauli weight 7 (claimed upper_bound)

witness operator (Pauli string, 7 qubits)

IIIIIIIXIIIIIIIIIIXIIIIIIIIIIXIIIIIIIIIIXIIIIIIIIIIXIIIIIIIIIIXIIIIIIIIIIXIII X: [7, 18, 29, 40, 51, 62, 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 8

qubit degrees S 8

trapping sets S (1,8)×77 (2,8)×77 (3,8)×77 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 77 (2,8): 77 (2,10): 77 (2,12): 693 (2,14): 231 (3,8): 77 (3,10): 231 (3,12): 2079 (3,14): 2926 (3,16): 6314 (3,18): 4928 (3,20): 693

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 77 cyclic shifts, a(x) = 1 + x¹¹ + x³⁸ + x³⁹ + x⁴⁹ + x⁵⁰, b(x) = x¹⁷ + x³⁹ + x⁴⁹ + x⁷¹ in F_2[x]/(x⁷⁷ - 1); a = g p, b = g q with p, q palindromic so that a b* + b a* = 0; k = deg gcd(a, b, x⁷⁷ - 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_77, 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

[[77,11,7]] — 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 = 77; a(x) = 1 + x^11 + x^38 + x^39 + x^49 + x^50; b(x) = x^17 + x^39 + x^49 + x^71. 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 [7, 18, 29, 40, 51, 62, 73], Z on [].

Stabilizer generators

generators 77 (max weight 8; 77 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 (77, Pauli strings on 77 qubits)

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symplectic rows (A | B) (77, sparse supports)

X: [0, 11, 38, 39, 49, 50] Z: [17, 39, 49, 71] X: [1, 12, 39, 40, 50, 51] Z: [18, 40, 50, 72] X: [2, 13, 40, 41, 51, 52] Z: [19, 41, 51, 73] X: [3, 14, 41, 42, 52, 53] Z: [20, 42, 52, 74] X: [4, 15, 42, 43, 53, 54] Z: [21, 43, 53, 75] X: [5, 16, 43, 44, 54, 55] Z: [22, 44, 54, 76] X: [6, 17, 44, 45, 55, 56] Z: [0, 23, 45, 55] X: [7, 18, 45, 46, 56, 57] Z: [1, 24, 46, 56] X: [8, 19, 46, 47, 57, 58] Z: [2, 25, 47, 57] X: [9, 20, 47, 48, 58, 59] Z: [3, 26, 48, 58] X: [10, 21, 48, 49, 59, 60] Z: [4, 27, 49, 59] X: [11, 22, 49, 50, 60, 61] Z: [5, 28, 50, 60] X: [12, 23, 50, 51, 61, 62] Z: [6, 29, 51, 61] X: [13, 24, 51, 52, 62, 63] Z: [7, 30, 52, 62] X: [14, 25, 52, 53, 63, 64] Z: [8, 31, 53, 63] X: [15, 26, 53, 54, 64, 65] Z: [9, 32, 54, 64] X: [16, 27, 54, 55, 65, 66] Z: [10, 33, 55, 65] X: [17, 28, 55, 56, 66, 67] Z: [11, 34, 56, 66] X: [18, 29, 56, 57, 67, 68] Z: [12, 35, 57, 67] X: [19, 30, 57, 58, 68, 69] Z: [13, 36, 58, 68] X: [20, 31, 58, 59, 69, 70] Z: [14, 37, 59, 69] X: [21, 32, 59, 60, 70, 71] Z: [15, 38, 60, 70] X: [22, 33, 60, 61, 71, 72] Z: [16, 39, 61, 71] X: [23, 34, 61, 62, 72, 73] Z: [17, 40, 62, 72] X: [24, 35, 62, 63, 73, 74] Z: [18, 41, 63, 73] X: [25, 36, 63, 64, 74, 75] Z: [19, 42, 64, 74] X: [26, 37, 64, 65, 75, 76] Z: [20, 43, 65, 75] X: [0, 27, 38, 65, 66, 76] Z: [21, 44, 66, 76] X: [0, 1, 28, 39, 66, 67] Z: [0, 22, 45, 67] X: [1, 2, 29, 40, 67, 68] Z: [1, 23, 46, 68] X: [2, 3, 30, 41, 68, 69] Z: [2, 24, 47, 69] X: [3, 4, 31, 42, 69, 70] Z: [3, 25, 48, 70] X: [4, 5, 32, 43, 70, 71] Z: [4, 26, 49, 71] X: [5, 6, 33, 44, 71, 72] Z: [5, 27, 50, 72] X: [6, 7, 34, 45, 72, 73] Z: [6, 28, 51, 73] X: [7, 8, 35, 46, 73, 74] Z: [7, 29, 52, 74] X: [8, 9, 36, 47, 74, 75] Z: [8, 30, 53, 75] X: [9, 10, 37, 48, 75, 76] Z: [9, 31, 54, 76] X: [0, 10, 11, 38, 49, 76] Z: [0, 10, 32, 55] X: [0, 1, 11, 12, 39, 50] Z: [1, 11, 33, 56] X: [1, 2, 12, 13, 40, 51] Z: [2, 12, 34, 57] X: [2, 3, 13, 14, 41, 52] Z: [3, 13, 35, 58] X: [3, 4, 14, 15, 42, 53] Z: [4, 14, 36, 59] X: [4, 5, 15, 16, 43, 54] Z: [5, 15, 37, 60] X: [5, 6, 16, 17, 44, 55] Z: [6, 16, 38, 61] X: [6, 7, 17, 18, 45, 56] Z: [7, 17, 39, 62] X: [7, 8, 18, 19, 46, 57] Z: [8, 18, 40, 63] X: [8, 9, 19, 20, 47, 58] Z: [9, 19, 41, 64] X: [9, 10, 20, 21, 48, 59] Z: [10, 20, 42, 65] X: [10, 11, 21, 22, 49, 60] Z: [11, 21, 43, 66] X: [11, 12, 22, 23, 50, 61] Z: [12, 22, 44, 67] X: [12, 13, 23, 24, 51, 62] Z: [13, 23, 45, 68] X: [13, 14, 24, 25, 52, 63] Z: [14, 24, 46, 69] X: [14, 15, 25, 26, 53, 64] Z: [15, 25, 47, 70] X: [15, 16, 26, 27, 54, 65] Z: [16, 26, 48, 71] X: [16, 17, 27, 28, 55, 66] Z: [17, 27, 49, 72] X: [17, 18, 28, 29, 56, 67] Z: [18, 28, 50, 73] X: [18, 19, 29, 30, 57, 68] Z: [19, 29, 51, 74] X: [19, 20, 30, 31, 58, 69] Z: [20, 30, 52, 75] X: [20, 21, 31, 32, 59, 70] Z: [21, 31, 53, 76] X: [21, 22, 32, 33, 60, 71] Z: [0, 22, 32, 54] X: [22, 23, 33, 34, 61, 72] Z: [1, 23, 33, 55] X: [23, 24, 34, 35, 62, 73] Z: [2, 24, 34, 56] X: [24, 25, 35, 36, 63, 74] Z: [3, 25, 35, 57] X: [25, 26, 36, 37, 64, 75] Z: [4, 26, 36, 58] X: [26, 27, 37, 38, 65, 76] Z: [5, 27, 37, 59] X: [0, 27, 28, 38, 39, 66] Z: [6, 28, 38, 60] X: [1, 28, 29, 39, 40, 67] Z: [7, 29, 39, 61] X: [2, 29, 30, 40, 41, 68] Z: [8, 30, 40, 62] X: [3, 30, 31, 41, 42, 69] Z: [9, 31, 41, 63] X: [4, 31, 32, 42, 43, 70] Z: [10, 32, 42, 64] X: [5, 32, 33, 43, 44, 71] Z: [11, 33, 43, 65] X: [6, 33, 34, 44, 45, 72] Z: [12, 34, 44, 66] X: [7, 34, 35, 45, 46, 73] Z: [13, 35, 45, 67] X: [8, 35, 36, 46, 47, 74] Z: [14, 36, 46, 68] X: [9, 36, 37, 47, 48, 75] Z: [15, 37, 47, 69] X: [10, 37, 38, 48, 49, 76] Z: [16, 38, 48, 70]

Code ID 77-11-7 · download JSON · raw on GitHub