Solution:

The code word is TIE.



Let us assume that the word is MOD. Now if you are told one character of this word, you would not be able to identify the number of vowels in the word. For example if you are told M, then the word can be MOD or AIM but they both have different number of vowels. If you are told about O, then there are two words MOD and OAT which again have different number of vowels. If you are told about D, there are two words again MOD and DUE which have different numbers of vowels again.



In such manner, all the words that have M, O or D in them can be excluded. Then only word that remains is TIE.

If you are told about T, there are two words OAT and TIE which both have two vowels.

If you are told about I, there are two words AIM and TIE which both have two vowels.

If you are told about E, there are two words DUE and TIE which both have two vowels.



Thus TIE is the code word.