A simple method based on Newton's identities and their extensions is presented for determining the actual minimum distance of cyclic codes. More significantly, it is shown that this method also provides a mechanism for generating the type of syndrome matrices needed by Feng and Tzeng's (see ibid., vol.40, p.1364-1374, Sept. 1994) new procedure for decoding cyclic and BCH codes up to their actual minimum distance. Two procedures for generating such matrices are given. With these procedures, we have generated syndrome matrices having only one class of conjugate syndromes on the minor diagonal for all binary cyclic codes of length n<63 and many codes of length 63/spl les/n/spl les/99. A listing of such syndrome matrices for selected codes of length n<63 is included. An interesting connection of the method presented to the shifting technique of van Lint (1986) and Wilson is also noted.