Paper 2023/1209

Infinite families of minimal binary codes via Krawtchouk polynomials

Xiaoni Du, Northwest Normal University, Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics
René Rodríguez, University of Primorska
Hao Wu, Northwest Normal University
Abstract

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, the general Maiorana-McFarland class. We employ a method first proposed by Ding et al. [7] to construct minimal codes violating the Ashikhmin-Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal codes. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. This study provides insights into the scope of this method.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Preprint.
Keywords
Minimal binary codesAshikhmin-Barg boundBoolean functionsMaiorana-McFarland classKrawtchouck polynomials.
Contact author(s)
ymldxn @ 126 com
rene7ca @ gmail com
nwnuwh @ 126 com
History
2023-08-10: approved
2023-08-09: received
See all versions
Short URL
https://ia.cr/2023/1209
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2023/1209,
      author = {Xiaoni Du and René Rodríguez and Hao Wu},
      title = {Infinite families of minimal binary codes via Krawtchouk polynomials},
      howpublished = {Cryptology {ePrint} Archive, Paper 2023/1209},
      year = {2023},
      url = {https://eprint.iacr.org/2023/1209}
}
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