Constructing Proper Codes for Error Detection

It is shown that for any alphabet size q and any code size M, there exist proper codes for error detection on a q-ary symmetric channel for all sufficiently large lengths. The stronger condition zero-strong proper code is defined. It is shown that such codes can only exist for q dividing M, and if this is the case they are shown to exist for sufficiently large lengths. The existence proofs are constructive.