Special subset sums: optimum

Let $S(A)$ represent the sum of elements in set $A$ of size $n$. We shall call it a special sum set if for any two non-empty disjoint subsets, $B$ and $C$, the following properties are true:

1. $S(B) \ne S(C)$; that is, sums of subsets cannot be equal.
2. If $B$ contains more elements than $C$ then $S(B) > S(C)$.

If $S(A)$ is minimised for a given $n$, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.

$$ \begin{aligned} n &= 1:{1}\ n &=2:{1, 2}\ n &=3:{2, 3, 4}\ n &=4:{3, 5, 6, 7}\ n &=5:{6, 9, 11, 12, 13} \end{aligned}

$$

It seems that for a given optimum set, $A = {a_1, a_2, \dots , a_n}$, the next optimum set is of the form $B = {b, a_1+b, a_2+b, ... ,a_n+b}$, where $b$ is the "middle" element on the previous row.

By applying this "rule" we would expect the optimum set for $n = 6$ to be $A = {11, 17, 20, 22, 23, 24}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = {11, 18, 19, 20, 22, 25}$, with $S(A) = 115$ and corresponding set string: $111819202225$.

Given that $A$ is an optimum special sum set for $n = 7$, find its set string.

NOTE: This problem is related to Problem 105 and Problem 106.