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

Abstract

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 \(\PSL_2(n)\).