Methods and systems for error detection and error correction in n-valued with n>2 data symbols are disclosed. N-valued check symbols are generated from data symbols in n-valued logic expressions using n-valued logic functions. N-valued Hamming codes are disclosed. Also disclosed is the generation of check symbols from data symbols in an n-valued expression wherein at least one check symbol is multiplied by a factor not equal to 0 or 1 in GF(n). Identifying n-valued symbols in error by check symbols and error correction by solving sets of independent n-valued equations are also disclosed. A method for introducing and removing annoyance errors is provided. Systems for error corrections in communication and data storage are also provided.