Extended Irreducible Binary Sextic Goppa Codes

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
$2^{n}+1$
.