Some Necessary Conditions for Codes to Be Good for Error Detection

Codes for error detection on a q-ary symmetric channel are studied. Whether a code is good or not for error detection (in the technical sense) depends on the structure of the code. For some combinations of the main parameters length, size, and minimum distance, all code are good and for some other combinations all are ugly (stronger than not good). The purpose of this paper is to give bounds on the parameters for codes that are not good for error detection. In particular, it is shown that if the minimum distance is below some bound, which depends on the length and size of the code as well as a lower bound on the number of codewords of minimum distance, then the code is ugly and, hence, not good for error detection.