Summary: Single error correcting-double error detecting (SEC-DED) codes have been successfully used in computer memories for reliability. In the present- day technology of very large scale integration storage arrays bit error correction as well as byte error detection/byte error correction become extremely important. We devise here classes of cyclic codes with generator of the form \((x^ b-1)\cdot p(x)\) for some suitable irreducible polynomial p(x) which provide 1) single byte error correction (SBEC), and 2) single error correction, double error detection, and byte error detection (SEC-DED-BED) codes. The codes have high information rate and are easy to implement. This general class of codes combines elegantly the results of several researchers and in some cases extends their results.