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The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass ofcyclic codes that have found numerous applications in error correction andnotably in quantum information processing. A subclass of attractive BCH codesis the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ withlength $q+1$, which are closely related to the action of the projective generallinear group of degree two on the projective line. This paper aims to studysome of the codes within this class and specifically narrow-sense antiprimitiveBCH codes (these codes are also linear complementary duals (LCD) codes thathave interesting practical recent applications in cryptography, among otherbenefits). We shall use tools and combine arguments from algebraic codingtheory, combinatorial designs, and group theory (group actions, representationtheory of finite groups, etc.) to investigate narrow-sense antiprimitive BCHCodes and extend results from the recent literature. Notably, the dimension,the minimum distance of some $q$-ary BCH codes with length $q+1$, and theirduals are determined in this paper. The dual codes of the narrow-senseantiprimitive 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 structuresinvariant 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 Wittspherical geometry designs is presented.  The BCH codes proposed in this paper are good candidates for permutationdecoding, as they have a relatively large group of automorphisms.

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and their Consequences