Geometric Obstructions and Sharp Genus Dependence in LCD Scalings

We study the least pole order of a rational function whose nonzero evaluations scale a prescribed code to a Euclidean LCD code. A residue-one differential yields an obstruction for odd-dimensional one-point AG codes in characteristic two: for C=CL(D,aQ) with 0≤ a<n and odd dimension, the threshold is at least $n+2g-a-1$. The obstruction uses alternating Gram matrices and does not require the scaled code to be self-orthogonal. For the dimension-three codes CL(D,(r+1)Q) on Hermitian curves over r², r=2ᵉ≥4, we obtain \[ n+2g-r-2≤τ_D(C;Q)≤ n+2g-3. \] Consequently the leading genus contribution is asymptotically sharp even in dimension three. On the elliptic curve y²+y=x³ over ₄, we prove the exact dimension-three threshold $6$ by an explicit multiplier. For repetition codes on the Hermitian family, the threshold equals the general interpolation bound $n+2g-1$. The upper bounds use information sets and hold for $q>3k$, and for $q>2k$ in characteristic two.