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=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 Fr2, r=2e≥4, we obtain n+2g−r−2≤τD(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 F4, 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.
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BibTeX
@misc{rahmatiasghar2026geometric,
title = {{Geometric Obstructions and Sharp Genus Dependence in LCD Scalings}},
author = {Rahim Rahmati-Asghar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04863},
eprint = {2611.04863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04863},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Obstructions and Sharp Genus Dependence in LCD Scalings
AU - Rahim Rahmati-Asghar
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04863
UR - https://arxiv.org/abs/2611.04863
AB - 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≤ 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≥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 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.
ER -