[[17,1,7]] · QEC Challenge

[[17,1,7]] d ≤stabilizer

kd²/n

2.882

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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 (2 Y factors; Hamming weight over 2n bits 9) (claimed upper_bound)

witness operator (Pauli string, 7 qubits)

XIIIZXIIIIZIYYZII X: [0, 5, 12, 13] Z: [4, 10, 12, 13, 14]

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)×17 (2,6)×34 (3,6)×153 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,8): 17 (2,6): 34 (2,8): 34 (2,10): 34 (2,12): 34 (3,6): 153 (3,8): 272 (3,10): 170 (3,12): 68 (3,14): 17

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction Cyclic stabilizer code, one-block circulant form: generator X^a Z^b with a(x) = x¹ + x¹⁶, b(x) = x³ + x⁵ + x⁶ + x¹¹ + x¹² + x¹⁴ in F_2[x]/(x¹⁷ - 1) and its n cyclic shifts; palindromic a, b make the shifts commute

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:quant-ph/9608006

date 2026-09-30

notes Known parameters, new generators: the parameters are those of the cyclic (GF(4)-additive) code of Calderbank-Rains-Shor-Sloane / codetables.de at n = 17, k = 1. Generators rebuilt by exhaustive search over palindromic circulant pairs; distance 7 is exact by enumeration of all Paulis of weight < 7 (see the note); filed as upper_bound because the board's certifier does not minimise Pauli weight. Not a Hadamard image of a CSS code (generators carry X and Z on overlapping supports with k = 1 at this n); dedup gate found no match. The generators filed here are the ones our search found; equivalence to the published cyclic code was not checked, so this is a submission with known parameters, credited to CRSS for the parameters.

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

[[17,1,7]] — cyclic stabilizer code, one-block circulant form X^a(x) Z^b(x), weight 8

Direction & hypothesis

A known-parameters submission for the general-stabilizer board. Cyclic quantum codes are additive cyclic codes over GF(4) (Calderbank, Rains, Shor, Sloane, arXiv:quant-ph/9608006); the parameters [[17,1,7]] are the cyclic-code entries of their tables and of codetables.de for n = 17, k = 1. This entry reproduces those parameters with generators found by our own search; equivalence to the published cyclic code was not checked, so it is filed as a submission with novelty: known_parameters.

What was searched

The code was rebuilt rather than transcribed: over F_2[x]/(x^17 - 1) we enumerated pairs of palindromic polynomials (a, b) of weight at most 6 (palindromic means a(x) = a(1/x)); for such pairs the n cyclic shifts of the generator X^a Z^b commute automatically, since a b* + b a* = 0. For each pair we computed k = n - rank of the symplectic matrix and the exact distance by enumerating every Pauli operator of weight below the target (vectorised over the 3^w patterns of each support). The lightest pair reaching the table distance is filed.

Evidence trail

Exact: no logical operator of Pauli weight below 7 exists (full enumeration of all

Paulis of weight <= 6: each either fails to commute with a generator or lies in the stabilizer group), and the JSON witness has weight 7. The board records stabilizer distances as upper bounds because its certifier does not minimise Pauli weight; the enumeration here is the exact statement.

qldpc submit's Pauli-weight RIS search (20,000 trials) also returned 7.

Dead ends

Weight-2 pairs give d = 1 (a = b); the first weight-4 pair reaching each n's table distance is the one filed. No weight-4 or weight-6 palindromic pair reaches d = 7 at n = 17: the best weight-4 pair (a = x + x^16, b = x^4 + x^13) stops at d = 5.

Tools

Claude Fable 5.1 in Claude Code; numpy for the enumeration; this repository's cli/qldpc.py and verify/ for the submission.

Reproduction

n = 17; a(x) = x^1 + x^16, b(x) = x^3 + x^5 + x^6 + x^11 + x^12 + x^14. Generator i (i = 0..n-1) is X on qubits {i + e mod n : e in supp a} and Z on {i + e mod n : e in supp b} (a qubit in both carries Y). The symplectic matrix is S = (circ(a) | circ(b)); k = n - rank_2(S) = 1.

Stabilizer generators

generators 17 (max weight 8; 17 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 (17, Pauli strings on 17 qubits)

IXIZIZZIIIIZZIZIX XIXIZIZZIIIIZZIZI IXIXIZIZZIIIIZZIZ ZIXIXIZIZZIIIIZZI IZIXIXIZIZZIIIIZZ ZIZIXIXIZIZZIIIIZ ZZIZIXIXIZIZZIIII IZZIZIXIXIZIZZIII IIZZIZIXIXIZIZZII IIIZZIZIXIXIZIZZI IIIIZZIZIXIXIZIZZ ZIIIIZZIZIXIXIZIZ ZZIIIIZZIZIXIXIZI IZZIIIIZZIZIXIXIZ ZIZZIIIIZZIZIXIXI IZIZZIIIIZZIZIXIX XIZIZZIIIIZZIZIXI

symplectic rows (A | B) (17, sparse supports)

X: [1, 16] Z: [3, 5, 6, 11, 12, 14] X: [0, 2] Z: [4, 6, 7, 12, 13, 15] X: [1, 3] Z: [5, 7, 8, 13, 14, 16] X: [2, 4] Z: [0, 6, 8, 9, 14, 15] X: [3, 5] Z: [1, 7, 9, 10, 15, 16] X: [4, 6] Z: [0, 2, 8, 10, 11, 16] X: [5, 7] Z: [0, 1, 3, 9, 11, 12] X: [6, 8] Z: [1, 2, 4, 10, 12, 13] X: [7, 9] Z: [2, 3, 5, 11, 13, 14] X: [8, 10] Z: [3, 4, 6, 12, 14, 15] X: [9, 11] Z: [4, 5, 7, 13, 15, 16] X: [10, 12] Z: [0, 5, 6, 8, 14, 16] X: [11, 13] Z: [0, 1, 6, 7, 9, 15] X: [12, 14] Z: [1, 2, 7, 8, 10, 16] X: [13, 15] Z: [0, 2, 3, 8, 9, 11] X: [14, 16] Z: [1, 3, 4, 9, 10, 12] X: [0, 15] Z: [2, 4, 5, 10, 11, 13]

Code ID 17-1-7 · download JSON · raw on GitHub