Let$n (>3)$be a prime number and${\mathbb {F}}_{2^{n}}$a finite field of$2^{n}$elements. Let$L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$be the support set and$g(x)$an irreducible polynomial of degree 6 over${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes$\Gamma (L, g)$of degree 6 and length$2^{n}+1$.

Twisted generalized Reed-Solomon (TGRS) codes are a family of codes that contains a large number of maximum distance separable (MDS) codes that are non-equivalent to generalized Reed-Solomon (GRS) codes. In this paper, we characterize a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS; give a sufficient and necessary condition that a TGRS code with two twists is self-dual, and present some constructions of self-dual TGRS codes. These self-dual codes are MDS, NMDS or 2-MDS. Furthermore, we study the non-GRS properties of TGRS codes with two twists and prove that these codes are non-GRS in most cases.