In this paper, we use extended Euclid’s algorithm to propose new decoding algorithms for two classes of maximum distance separable (MDS) twisted generalized Reed-Solomon (TGRS) codes of parameters$[n, n-t, t+1]$over$\Bbb F_{q}$. For even t, the algorithms can correct$\frac {t}{2}$errors with time complexity$O(qn)$. Moreover, we also give a new decoding algorithm for a class of twisted Goppa codes. For even degree t of a Goppa polynomial, it can also correct$\frac {t}{2}$errors, which generalizes a$\lfloor \frac {t-1}{2}\rfloor $-error-correcting decoding algorithm by Sui and Yue (2023).

Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let$\Bbb F_{q}$be a finite field with$q=2^{l}$elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism$A \in PGL_{2}(\Bbb F_{q})$. Moreover, we provide some examples to support our findings.