Summary: Orthomorphisms have important application in the design of cryptosystems. Linear orthomorphisms on GF\( (2^n)^m\) can be used to design the important linear part \(p\)-permutation in block cipher. This paper generalizes the orthomorphisms on GF\( (2)^m\) and the perfect balance on GF\( (2)^m\) to the ones on GF\( (2^n)^m\) by the first time, and proves that orthomorphisms on GF\( (2^n)^m\) are perfectly balanced, then proves a new theorem about GF\( (q)[x]\), and then designs a structure and a counting method of linear orthomorphisms on GF\( (2^n)^m\). At last it gives the formula for counting all the linear orthomorphisms on GF\( (2^n)^m\) by the new theorem about GF\( (q)[x]\).