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$A(2)$ and 
$A(3)$
Let 
$ \mathcal{X} $ be a curve over 
${ \mathbb{F} }_{q} $ and let 
$N( \mathcal{X} )$, 
$g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant 
$A(q)$ is defined by 
$A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on 
$A(2)$ and 
$A(3)$.
A lower bound on the number of points that can be placed in a square of side 
$\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.
