The projective special linear group PSL2(n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... , n - 1, ∞). It is known that the extended odd-like quadratic residue codes are invariant under PSL2(n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n = 1 (mod 4), an infinite family of 3-designs for primes n = 3 (mod 4). To construct more t-designs with t ∈ (2, 31, one would search for other extended cyclic codes over finite fields that are invariant under the action of PSL2(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 PSL2(n).

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.