Finding numbers by given XOR values. Given XOR values of 3 indices how can we find the numbers?
Like say if I have indices from 1 to 7, how can I find the numbers by given XOR values?
I have:


*

*$X_{1} \oplus X_{3} \oplus X_{5}=V_1$

*$X_{1} \oplus X_{3} \oplus X_{6}=V_2$

*$X_{1} \oplus X_{4} \oplus X_{6}=V_3$

*$X_{2} \oplus X_{4} \oplus X_{6}=V_4$

*$X_{2} \oplus X_{4} \oplus X_{7}=V_5$

*$X_{2} \oplus X_{5} \oplus X_{7}=V_6$

*$X_{3} \oplus X_{5} \oplus X_{7}=V_7$
How can I find any $X_{i}$ from the above data? Is there any pattern?

Build the matrix $A$ in which the entry $A_{i,j}$ is $1$ if in the $i$th equation there is $X_j$ appearing on the left hand side and $0$ otherwise.
The XOR system is equivalent to the following system of linear equations over the field $\mathbb F_2$:
$$
\left(
\begin{array}{ccccccc}
 1 & 0 & 1 & 0 & 1 & 0 & 0 \\
 1 & 0 & 1 & 0 & 0 & 1 & 0 \\
 1 & 0 & 0 & 1 & 0 & 1 & 0 \\
 0 & 1 & 0 & 1 & 0 & 1 & 0 \\
 0 & 1 & 0 & 1 & 0 & 0 & 1 \\
 0 & 1 & 0 & 0 & 1 & 0 & 1 \\
 0 & 0 & 1 & 0 & 1 & 0 & 1 \\
\end{array}
\right) \cdot
\begin{pmatrix} X_1\\X_2\\X_3\\X_4\\X_5\\X_6\\X_7 \end{pmatrix}
=
\begin{pmatrix} V_1\\V_2\\V_3\\V_4\\V_5\\V_6\\V_7 \end{pmatrix}
$$
The matrix on the left is the matrix $A$ that I made you build at the beginning.
The system is certainly solvable if the matrix $A$ is invertible in $\mathbb F_2$.
In our case, $\det(A)=-3$, so it is possible to solve the system.
This is the inverse matrix:
$$
\left(
\begin{array}{ccccccc}
 1 & 1 & 1 & 0 & 1 & 1 & 0 \\
 1 & 1 & 0 & 1 & 1 & 1 & 0 \\
 1 & 1 & 0 & 1 & 1 & 0 & 1 \\
 1 & 0 & 1 & 1 & 1 & 0 & 1 \\
 1 & 0 & 1 & 1 & 0 & 1 & 1 \\
 0 & 1 & 1 & 1 & 0 & 1 & 1 \\
 0 & 1 & 1 & 0 & 1 & 1 & 1 \\
\end{array}
\right) .
$$
To compute $X_i$, read the $i$th row and look for the columns where $1$ is present. Those are the indices of your original equations that you have to XOR together. For instance, for $X_1$ you have to XOR together
$$
X_1 = V_1 \oplus V_2 \oplus V_3 \oplus V_5 \oplus V_6 .
$$
