15. A binary code is a string of digits $x _ { 1 } x _ { 2 } \cdots x _ { n }$ ($n \in \mathbb{N} ^ { * }$) composed of 0 and 1, where $x _ { k }$ ($k = 1,2, \cdots, n$) is called the $k$-th bit code element. Binary codes are commonly used in communication, but during the communication process\\
\includegraphics[max width=\textwidth, alt={}]{af8058d3-52ab-42d0-ba33-0c5c3a276f48-02_410_273_1758_1564}code element errors sometimes occur (that is, a code element changes from 0 to 1, or from 1 to 0).

A certain type of binary code $x _ { 1 } x _ { 2 } \cdots x _ { 7 }$ satisfies the following system of check equations: $\left\{ \begin{array} { l } x _ { 4 } \oplus x _ { 5 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 2 } \oplus x _ { 3 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 1 } \oplus x _ { 3 } \oplus x _ { 5 } \oplus x _ { 7 } = 0 , \end{array} \right.$\\
where the operation $\oplus$ is defined as: $0 \oplus 0 = 0$, $0 \oplus 1 = 1$, $1 \oplus 0 = 1$, $1 \oplus 1 = 0$.\\
It is now known that such a binary code became 1101101 after a code element error occurred at the $k$-th position during transmission. Using the above system of check equations, we can determine that $k$ equals $\_\_\_\_$.