All binary linear codes that are invariant under $\PSL_2(n)$

The projective special linear group $\PSL_2(n)$ is $2$-transitive for all primes $n$ and $3$-homogeneous for $n \equiv 3 \pmod{4}$ on the set $\{0,1, \cdots, n-1, \infty\}$. It is known that the extended odd-like quadratic residue codes are invariant under $\PSL_2(n)$. Hence, the extended quadratic residue codes hold an infinite family of $2$-designs for primes $n \equiv 1 \pmod{4}$, an infinite family of $3$-designs for primes $n \equiv 3 \pmod{4}$. To construct more $t$-designs with $t \in \{2, 3\}$, one would search for other extended cyclic codes over finite fields that are invariant under the action of $\PSL_2(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under $\PSL_2(n)$.