长度为512的四阶Reed-Muller码的权值分布

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-03-07 DOI:10.1007/s10623-025-01602-2

Miroslav Markov, Yuri Borissov

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引用次数: 0

摘要

我们通过结合d.v. Sarwate 1973年博士论文中描述的方法和布尔函数仿射等价分类的新结果来计算二进制Reed-Muller码\({\mathcal {R}} (4,9)\)的权重分布。更具体地说，为了解决这个问题，例如，在MacWilliams和Sloane的书中，我们采用了一种基于Ph. Langevin和G. Leander在八个变量中布尔四次形式分类的增强方法，以及V. Gillot和Ph. Langevin最近获得的商空间分类\({\mathcal {R}} (4,7)/{\mathcal {R}} (2,7)\)的结果。

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The weight distribution of the fourth-order Reed–Muller code of length 512

We compute the weight distribution of the binary Reed–Muller code \({\mathcal {R}} (4,9)\) by combining the methodology described in D. V. Sarwate’s Ph.D. thesis from 1973 with newer results on the affine equivalence classification of Boolean functions. More specifically, to address this problem posed, e.g., in the book of MacWilliams and Sloane, we apply an enhanced approach based on the classification of Boolean quartic forms in eight variables due to Ph. Langevin and G. Leander, and the recent results on classification of the quotient space \({\mathcal {R}} (4,7)/{\mathcal {R}} (2,7)\) obtained by V. Gillot and Ph. Langevin.

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.