t-random-error correcting and all-unidirectional-error detecting (t-EC/AUED) codes obtained by appending a suffix to each codeword of a linear block code are presented. Although this technique has been previously used to construct such codes, a different approach is proposed that involves starting with a nonoptimal t-EC linear code to obtain more efficient t-EC/AUED codes. Specifically, t-EC/AUED codes constructed from an even-weight t-EC linear code are shown to require a shorter suffix than when a general t-EC linear code is used. The t-EC/AUED codes obtained in this manner are thoroughly compared to the best known codes, and are shown to require fewer bits in almost all cases. >