I have been reading the paper Monotonic optimization based decoding for linear codes. In this paper the author show that modulo 2 operation can be rewritten as cos function.
Particularly, an $\left( {n,k} \right)$ with the corresponding generator matrix $G$ of size $n \times k$ is considered
$G = \left[ {\begin{array}{*{20}{c}} {{g_1}}\\ {{g_2}}\\ {...}\\ {{g_n}} \end{array}} \right]$ each row is ${g_i} \in {\left\{ {0,1} \right\}^k}$
Let $u, x$ be the column vectors of the input bit sequence/message and the corresponding encoder output codeword. This will give the relation $x = Gu\left( {\bmod \,\,2} \right)$. The author then show this without any proof or intuition explain
$Gu\left( {\bmod \,\,2} \right) = \frac{{{1_n} - \cos \left( {\pi Gu} \right)}}{2}$ where $\cos \left( {\pi Gu} \right): = {\left( {\cos \left( {\pi {g_1}u} \right),\cos \left( {\pi {g_2}u} \right),\cos \left( {\pi {g_3}u} \right),...,\cos \left( {\pi {g_n}u} \right)} \right)^T}$
How can modulo 2 be express as cos function (for optimization latter in the continuous space) ?
Please help me understand this
Thank you very much
$$\frac{1-\cos0}2 = 0\\ \frac{1-\cos\pi}2 = 1\\ \frac{1-\cos2\pi}2 = 0\\ \frac{1-\cos3\pi}2 = 1\\ \vdots$$