1. A method for time or transform domain decoding of RS coded received messages which corrects both errors and erasures in decoding of RS coded messages by the use of the Euclidean algorithm without separate computations of the errata locator polynomial .tau.(x) and the errata evaluator polynomial A(x) comprising the steps of (a) computing the transform of the received m-bit vector over the Galois Field GF(2.sup.m) from Eq. (2) for the initial syndrome computation, then calculate the erasure locator polynomial .LAMBDA.(x) and defined deg {.lambda.(x)}=v, (b) computing the Forney syndrome polynomial T(x) from Eq. (7), (c) applying the Euclidean algorithm to x.sup.d-1 and T(x) as given by Eq. (7) to determine the errata locator polynomial .tau.(x) and errata evaluator polynomial A(x), using as initial values of the Euclidean algorithm .tau..sub.0 (x)=.LAMBDA.(x), .tau..sub.-1 (x)=0, R.sub.-1 (x)=x.sup.d-1 and R.sub.0 (x)=T(x), wherein the recursion in Eq. (9) for R.sub.s (x) and .tau..sub.s (x) terminates when deg {R.sub.s (x)}.ltoreq. (d+v-3)/2 for the first time for some value s=s', and then compute .tau.(x) and A(x) from Eqs. (10a), and (10b), and (d) for time domain decoding, computing the error and erasure locations by finding the roots of the errata location polynomial .tau.(x) by Eq. (4c), or for transform domain decoding, determining the error and erasure locations by subjecting the errata locations polynomial .tau.(x) and the syndrome S(x) from the syndrome computaton in step (a) to a transform error pattern calculation which produces extended syndromes and inverting the sequence of the error pattern thus determined.