What are the sufficient conditions for weighted binary code to be self compliementary?
You haven't defined your terms, so let me quickly define them:


A weighted binary code $abcd$ is the mapping $C\colon \{0,1\}^4 \to \mathbb{N}$ given by $$C(xyzw) = ax + by + cz + dw.$$
A code $C$ is legal (my term) if $0,\ldots,9$ are all in the range of $C$.
A code $C$ is self-complementing if whenever $C(xyzw) \in \{0,\ldots,9\}$, we have
$$ C(\bar{x}\bar{y}\bar{z}\bar{w}) = 9-C(xyzw), $$
where $\bar{x}$ is the negation of $x$.


Since $\bar{x} = 1-x$, we have
$$
C(\bar{x}\bar{y}\bar{z}\bar{w}) = a(1-x)+b(1-y)+c(1-z)+d(1-w) = a+b+c+d - C(xyzw).
$$
Therefore a legal code is self-complementing iff $a+b+c+d=9$.