Abstarct: Designing proper codes, in both theoretical and applied perspectives, is considered as an important problem in the coding theory. Since appearance of coding theory, many researchers has tried so much and thereby has generated various interesting kinds of these codes. In this thesis, we want to study the existence or nonexistence of perfect codes in discrete simplexes and multi-set codes in permutation channels. First, it is discussed about the introduction of coding theory and required definitions for the remaining chapters. Then perfect codes in discrete simplex are analyzed in different dimensions and we conclude that perfect codes always exists in the case of binary alphabet, whereas in the case of ternary alphabet we have perfect codes just in some special cases. Actually we show that perfect codes in 1-simplex exist for any , the 2-simplex admits a perfect code if and only if , while there are no perfect codes in higher dimensional simplexes. In other words, perfect codes exist only over binary and ternary alphabets. Then we investigate multi-set codes in permutation channels and errors including deletion, insertion and substitution in permutation channels. Finally, existence of perfect codes in and lattices are investigated.