I have these set of vectors;
$$(1,-1,-1,1)$$ $$(1,-1,1,-1)$$ $$(1,1,-1,-1) $$ $$(A,B,C,D)$$
The conditions on A,B,C,D are that they have to be either be 1 or -1, these elements have to add up to 0 and they have to be orthogonal to all the above vectors. Can someone show me a method of finding A,B,C and D?
Two contrasts (i.e. vectors with $\pm1$ that sum up to zero) are orthogonal if their inner product (sumproduct of their coordinates) is equal to zero. Thus the vector $(A,B,C,D)$ is orthogonal to a vector of the form $(c_1, c_2, c_3,c_4)$ where $c_i=\pm1$ if $$c_1A+c_2B+c_3C+c_4D=0$$ In the case above this yields $$\begin{cases}A-B-C+D=0\\A-B+C-D=0\\A+B-C-D=0\\A+B+C+D=0\end{cases}$$ where the last equation ensures that the vector $(A,B,C,D)$ is also a contrast, i.e. it's entries sum up to zero.
If you sum up all equations, you have that $$4A=0 \implies A=0$$ which means that there is no solution with the given criteria (entries equal to $\pm1$ and orthogonal to the above vectors).