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Binary linear codes with at most three weights from cyclotomic mappings.

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Title: Binary linear codes with at most three weights from cyclotomic mappings.
Authors: Cui, Heming1 (AUTHOR) s23090029@s.upc.edu.cn, Zhao, Xubo1 (AUTHOR) zhaoxubo@upc.edu.cn, Wang, Qiang2 (AUTHOR) wang@math.carleton.ca, Li, Xiaoping1 (AUTHOR) xplteaa@163.com, Yan, Tongjiang1,3 (AUTHOR) yantoji@163.com
Source: Finite Fields & Their Applications. Jun2026, Vol. 112, pN.PAG-N.PAG. 1p.
Subjects: Linear codes, Cyclotomic fields, Message authentication codes, Cryptography
Abstract: Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24] , Wu et al. (2019) [25] , and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J , whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24] , Wu et al. (2019) [25] , and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J , whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. [ABSTRACT FROM AUTHOR]
ISSN:10715797
DOI:10.1016/j.ffa.2026.102798