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The projective special linear group $\PSL_2(n)$ is $2$-transitive for allprimes $n$ 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 quadraticresidue codes are invariant under $\PSL_2(n)$. Hence, the extended quadraticresidue 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 otherextended 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 extendedquadratic residue binary codes are the only nontrivial extended binary cycliccodes that are invariant under $\PSL_2(n)$.

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