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 },
 "family": "generalized-bicycle",
 "provenance": {
  "authors": [
   "@vprusso"
  ],
  "model": "Claude Opus 5",
  "novelty": "unknown",
  "origin": "submission",
  "date": "2026-10-13",
  "construction": "Cyclic generalized-bicycle code on the ring x^257 - 1 with H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T]. Both a and b lie in the ideal generated by a divisor of x^257 - 1 of degree 16, which fixes k = 2 deg gcd(a, b, x^257 - 1) = 32.",
  "notes": "m = 257 is prime with ord_2(m) = 16, so x^257 - 1 factors as (x + 1) times sixteen distinct degree-16 minimal polynomials, and the only proper divisor of m is 1, which makes every quotient fold vacuous. The divisor is one of those degree-16 factors, chosen among the sixteen to maximise the longest run of consecutive residues it leaves out. The single-block logicals form the cyclic code of dimension 16 whose zeros are the 241 residues the divisor leaves out, and the Bose-Chaudhuri-Hocquenghem bound on that run puts the lightest single-block logical well above this claim; a direct circulant search at 12 seeds of 600,000 trials finds nothing below 72. Draws with two even weights are rejected, since (x + 1) divides a binary polynomial exactly when its weight is even and would add the root-1 coset to the common gcd. The distance is a witness-backed upper bound from a 24-seed gate at 2,000,000 information-set trials per seed; the orbit fold over every equal-orbit partition of the qubits returns nothing lighter."
 }
}