Decoding Algorithms of Twisted GRS Codes and Twisted Goppa Codes

Abstract

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).