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

Qi Liu; Cunsheng Ding; Sihem MesnagerChunming TangVladimir D. Tonchev

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

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

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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 ${mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. 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 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.

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《IEEE Transactions on Information Theory》 |2022年第5期|3096-3107|共12页
作者
Qi Liu; Cunsheng Ding; Sihem MesnagerChunming TangVladimir D. Tonchev;
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作者单位

School of Mathematics and Information, China West Normal University, Nanchong, China;

The Hong Kong University of Science and Technology, Hong Kong, China;

University of Paris VIII, Saint-Denis, FranceSchool of Mathematics and Information, China West Normal University, Sichuan, Nanchong, ChinaMichigan Technological University, Houghton, MI, USA;

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Codes; Linear codes; Geometry; Liquid crystal displays; Generators; Decoding; Codecs;

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

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