The class of codes discussed in this paper has the property that its error-correction capability is described in terms of correcting errors in specific digits of a code word even though other digits in the code may be decoded incorrectly. To each digit of the code words is assigned an error protection levelf_{i}. Then, ifferrors occur in the reception of a code word, all digits which have protectionf_{i}greater than or equal tofwill be decoded correctly even though the entire code word may not be decoded correctly. Methods for synthesizing these codes are described and illustrated by examples. One method of synthesis involves combining the parity check matrices of two or more ordinary random error-correcting codes to form the parity check matrix of the new code. A decoding algorithm based upon the decoding algorithms of the component codes is presented. A second method of code generation is described which follows from the observation that for a linear code, the columns of the parity check matrix corresponding to the check positions must span the column space of the matrix. Upper and lower bounds are derived for the number of check digits required for such codes. The lower bound is based upon counting the number of unique syndromes required for a specified error-correction capability. The upper bound is the result of a constructive procedure for forming the parity check matrices of these codes. Tables of numerical values for the upper and lower bounds are presented.