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The class of linear codes introduced by Goppa are noncyclic in general. The only Goppa codes known to be cyclic are the Bose--Chaudhuri-Hocquengbem (BCH) codes which are a special class of Goppa codes. Recently, Berlekamp and Moreno showed that certain double-error-correcting binary Goppa codes become cyclic when extended by an overall parity check. In this correspondence, results of a further investigation on extending Goppa codes to cyclic codes are presented. It is shown that a large class of multiple-error-correcting -ary Goppa codes also become cyclic when extended by an overall parity check. These Goppa codes are found to be reversible. The Goppa codes considered in this correspondence consist of two subclasses that, after extension, give rise to the two subclasses of reversible cyclic codes of primitive and nonprimitive length, respectively. These cyclic codes are noted to include, respectively, the expurgated Melas codes and the Zetterberg codes as special cases.