To explain how addition and multiplication works in Galois fields, almost all the resources use the example of smallest finite field $\mathrm{GF}(2)$ that has elements $\{0,1\}$.
How can we define these properties for a $\mathrm{GF}(2)$ with elements $\{+1,-1\}$.
Can someone please explain the following statement:
Let $u$ be in $\mathrm{GF}(2)$ with the elements $\{+1,-1\}$, where $+1$ is the "null" element under the $\oplus$ addition.
What are the implications of this.
Reference: Hagenauer, J., "The exit chart - introduction to extrinsic information transfer in iterative processing," in Signal Processing Conference, 2004 12th European , vol., no., pp.1541-1548, 6-10 Sept. 2004
Thank you
If you use $+1$ as the additive identity, everything else is determined. Here are the operation tables.
$$\begin{array}{c|cc} \oplus&+1&-1\\ \hline +1&+1&-1 \\ -1&-1&+1\end{array}$$
$$\begin{array}{c|cc} \odot&+1&-1\\ \hline +1&+1&+1 \\ -1&+1&-1\end{array}$$