Covering finite fields by translates of multiplicative subgroups

Let $G_d$ be the subgroup of index $d>1$ in ${\mathbb{F}}_q^{\times }$, and put $G_d^0=G_d\cup \{0\}$. For a covering $A+G_d^0={\mathbb{F}}_q$ with $\mathrm{\#}A=s$, we prove $q\le d^{8s\mathrm{/}d}$ by shifting the roots of a Bonferroni polynomial. Although a uniform bound in $s$ alone must allow exponential growth, our estimate confines such growth to bounded indices and gives polynomial bounds when $s\mathrm{/}d$ is bounded. A cubic specialization gives bounds of degree at most eight for $s\le 2d$. For shift sets of logarithmic size, higher moment estimates give a uniform covering-probability formula, including the critical window. Together with the index restriction, this determines the covering indices of almost every shift set of size $\lceil k\log q\rceil$ for each fixed integer $k\ge 2$. A subfield construction attains the order of the uniform restriction on the index. We determine its covering number exactly and prove minimality under subgroup inclusion. We also characterize equality in the elementary counting bound by subfield cosets. Fixing a subfield as the shift set in a quadratic extension leads to a sharper projection bound, attained by an infinite family with unique representations outside the shift subfield.