Perfect Nonbinary AN Codes with Distance Three

It has been proved that the perfect binary and ternary single error correcting AN codes exist. In this paper, perfect radix- r single error correcting codes of type I and II are defined as generalized versions of perfect binary codes, and a general theory is developed for the existence of perfect single error correcting AN codes by using number theoretic concepts. This leads us to the followings: (1) There do not exist perfect radix 4 k 2 codes of type I and perfect radix k 2 codes of type II. (2) In the cases of radix 4, 5, 8 and 9, no perfect code of either type exists. (3) There exist perfect codes with radices 6 and 7.