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.
翻译:Bose-Chaudhuri- HOCIGNGH (BCH) 代码是一个研究周经的亚类循环代码, 它在错误校正和量信息处理中发现许多应用。 一个有吸引力的 BCH 代码子类是Galois 字段的狭义 BCH代码 $\ mathrm{GF}(q) 美元+1美元, 与投影线上的投影通用双线组的动作密切相关。 本文旨在研究该类中的某些代码, 特别是狭义的反皮价BCH代码( 这些代码也是线性双向的双向双向双向双向双向双向双向的代码 。 我们使用工具, 并结合变数理论( 组动作, 定数组的演示理论理论, 等) 以小于直线的 BCH 代码, 以近文中的平面值平面的平面读数为基数 。 直径, 平面的平面的平面的平面的平面的平面的平面代码 。