Smallest set reaching 1 to 40 by adding and subtracting

Find a set of integers of smallest possible size such that every whole number from $1$ to $40$ can be written as a signed sum of set elements, using each element at most once (with $+$, $-$ or not at all). Show that no smaller set works.

$\{1,3,9,27\}$, four elements. Each element has three choices, so $k$ elements produce at most $3^k$ signed sums, symmetric about $0$, giving at most $(3^k-1)/2$ positive values; $k=3$ gives only $13$. Balanced ternary shows $1,3,9,27$ reach every integer from $1$ to $40$.