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 "distance": {
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  "X": {
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    "tool": "research/kit/surrogate.lightest_logical",
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 "provenance": {
  "authors": [
   "@vprusso"
  ],
  "construction": "Cyclic generalized bicycle at m=165 = 3 x 5 x 11, with ord_2(m)=20, so x^165 - 1 factors over GF(2) into 14 cyclotomic cosets, 1 of size 1, 1 of size 2, 3 of size 4, 3 of size 10, 6 of size 20. The divisor g is assembled from coset minimal polynomials to degree 25, which fixes k = 2 deg g = 50 by construction, and (a,b) was drawn from the ideal (g) by information-set decoding over sparse supports, keeping weight 10 and weight 6 in the two blocks. Both weights are even, so x+1 divides the gcd; x+1 is x^1 - 1, so the only block logical this forces has weight m and is far above the claim. m is composite, so the quotient fold acts at t=3, t=5, t=11, t=15, t=33 and t=55; a fold over every equal-orbit partition of the qubits returns nothing lighter than 42, clear of the claim. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 330 qubits at uniform row weight 16, with a = [4, 37, 50, 65, 82, 103, 107, 116, 130, 134] and b = [11, 76, 102, 106, 126, 135].",
  "references": [],
  "date": "2026-10-15",
  "notes": "",
  "model": "Claude Opus 5"
 },
 "family": "generalized-bicycle"
}