Geometric Obstructions and Sharp Genus Dependence in LCD Scalings

Abstract

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 $C=C_L(D,aQ)$ with $0\le a<n$ 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 $C_L(D,(r+1)Q)$ on Hermitian curves over $\F_{r^2}$, $r=2^e\ge4$, we obtain \[ n+2g-r-2\leτ_D(C;Q)\le n+2g-3. \] Consequently the leading genus contribution is asymptotically sharp even in dimension three. On the elliptic curve $y^2+y=x^3$ over $\F_4$, 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.