  The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of
cyclic codes that have found numerous applications in error correction and
notably in quantum information processing. A subclass of attractive BCH codes
is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with
length $q+1$, which are closely related to the action of the projective general
linear group of degree two on the projective line. This paper aims to study
some of the codes within this class and specifically narrow-sense antiprimitive
BCH codes (these codes are also linear complementary duals (LCD) codes that
have interesting practical recent applications in cryptography, among other
benefits). We shall use tools and combine arguments from algebraic coding
theory, combinatorial designs, and group theory (group actions, representation
theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH
Codes and extend results from the recent literature. Notably, the dimension,
the minimum distance of some $q$-ary BCH codes with length $q+1$, and their
duals are determined in this paper. The dual codes of the narrow-sense
antiprimitive BCH codes derived in this paper include almost MDS codes.
Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over
$\mathrm{GF} (p^h)$ is completed.
  As an application of this result, the $p$-ranks of all incidence structures
invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$
are determined.
  Furthermore, infinite families of narrow-sense BCH codes admitting a
$3$-transitive automorphism group are obtained.
  Via these BCH codes, a coding-theory approach to constructing the Witt
spherical geometry designs is presented.
  The BCH codes proposed in this paper are good candidates for permutation
decoding, as they have a relatively large group of automorphisms.
