Optimal Binary Constant Weight Codes and Affine Linear Groups over Finite Fields

Let $\text{AGL}(1,\Bbb F_q)$ be the affine linear group of dimension $1$ over a finite field $\Bbb F_q$. $\text{AGL}(1,\Bbb F_q)$ acts sharply 2-transitively on $\Bbb F_q$. Given $S<\text{AGL}(1,\Bbb F_q)$ and an integer $k$ with $1\le k\le q$, does there exist a subset $B\subset\Bbb F_q$ with $|B|=k$ such that $S=\text{AGL}(1,\Bbb F_q)_B$? ($\text{AGL}(1,\Bbb F_q)_B=\{\sigma\in\text{AGL}(1,\Bbb F_q):\sigma(B)=B\}$ is the stabilizer of $B$ in $\text{AGL}(1,\Bbb F_q)$.) We derive a sum that holds the answer to this question. This result determines all possible parameters of binary constant weight codes that are constructed from the action of $\text{AGL}(1,\Bbb F_q)$ on $\Bbb F_q$ to meet the Johnson bound. Consequently, the values of the function $A_2(n,d,w)$ are determined for many parameters, where $A_2(n,d,w)$ is the maximum number of codewords in a binary constant weight code of length $n$, weight $w$ and minimum distance $\ge d$.