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

Abstract

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. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field 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. Despite its interest, not much is known about this class of BCH codes. 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 narrowsense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of PGL(2, p m )-invariant codes over 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 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 ⋆ C.

An infinite family of antiprimitive cyclic codes supporting Steiner systems $S(3,8, 7^m+1)$

“…A major way to construct combinatorial t-designs with linear codes over finite fields is the use of linear codes with t-transitive or t-homogeneous automorphism groups (see [10,Theorem 4.18]) and some combinatorial t-designs (see, for example, [7]) were obtained by this way. Very recently, Liu et al [28] obtained some 3-transitive automorphism groups from a class of BCH codes and derived some combinatorial 3-designs with this way. Another major way to construct t-designs with linear codes is the use of the Assmus-Mattson Theorem (AM Theorem for short) in [10,Theorem 4.14] and the generalized version of the AM Theorem in [22], which was recently employed to construct a number of t-designs (see, for example, [10], [24], [25]).…”

“…Specifically, the complements of the supports of the minimum weight codewords in C m form a Steiner system S(3, 8, 7 m + 1). Using the similar method of this paper and [28],…”

“…In this subsection, we will show that the cyclic code C m and its dual C ⊥ m are invariant under group actions of certain permutation groups which are 3-transitive, i.e., the automorphism groups of those code are 3-transitive. To this end, we use the similar method in Liu et al [28].…”

“…Lemma 11. [28] Let symbols and notation be the same as before. Let q = 7 m with m ≥ 2 being a positive integer.…”