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Binary linear codes with few weights from Boolean functions

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Abstract

Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de, and the duals of some of them are distance-optimal with respect to the sphere packing bound.

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Acknowledgements

The authors sincerely thank the editors and the reviewers for their helpful comments and valuable suggestions, which have improved the presentation of this paper. This work was partially supported by The National Natural Science Foundation of China under Grant Numbers 11971156 and 12001175, 61977021 and The Hubei Province Science and Technology Innovation Major Project under Grant Number 2019ACA144.

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Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics,Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Xiaoqiang Wang & Dabin Zheng

  2. School of Computer Science and Information Engineering, Hubei University, Wuhan, 430062, China

    Yan Zhang

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  1. Xiaoqiang Wang
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  2. Dabin Zheng
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  3. Yan Zhang
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Correspondence to Dabin Zheng.

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Communicated by C. Carlet.

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Wang, X., Zheng, D. & Zhang, Y. Binary linear codes with few weights from Boolean functions. Des. Codes Cryptogr. 89, 2009–2030 (2021). https://doi.org/10.1007/s10623-021-00898-0

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