We study the problem of maximizing the size of a ternary block code with given length and minimum Hamming distance. The problem is further restricted in two different ways. Either we require all codewords to have constant Hamming weight, or we require all codewords to have constant composition. We let the alphabet consist of the symbols zero, one and two. In a constant-composition code the number of zeros, the number of ones and the number of twos in each codeword are fixed, while in a constant-weight code only the total number of ones and twos in each codeword is fixed.

For both code classes several upper bounds are presented. We give a number of constructions of codes that meet the upper bounds. A construction of perfect ternary constant-weight codes with minimum distance three is presented. We have also compiled tables of the best possible upper and lower bounds on the code size.