A note on the Tu-Deng conjecture

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• 作者：Kaimin Cheng ; Shaofang Hong ; Yuanming Zhong
• 关键词：2 ; adic valuation ; Tu ; Deng conjecture ; weight
• 刊名：Journal of Systems Science and Complexity
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：28
• 期：3
• 页码：702-724
• 全文大小：270 KB
• 参考文献：[1]Tu Z and Deng Y, A conjecture about binary strings and its applications on constructing Boolean function with optimal algebraic immunity, Des. Codes Cryptogr., 2011, 60: 1-4.View Article MATH MathSciNet
  [2]Tu Z and Deng Y, Boolean functions with algebraic immunity one, Journal of Systems Science and Mathematical Sciences, 2011, 31(5): 512-18.MATH MathSciNet
  
• 作者单位：Kaimin Cheng (1) (2)
  Shaofang Hong (1)
  Yuanming Zhong (1)
  
  1. Mathematical College, Sichuan University, Chengdu, 610064, China
  2. Department of Mathematics, Sichuan University Jinjiang College, Pengshan, 620860, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Systems Theory and Control
  Applied Mathematics and Computational Methods of Engineering
  Operations Research/Decision Theory
  Probability Theory and Stochastic Processes
  
• 出版者：Academy of Mathematics and Systems Science, Chinese Academy of Sciences, co-published with Springer
• ISSN：1559-7067

文摘

Let k be a positive integer. For any positive integer x = Σ _i=0 ?/sup> x _i 2 i , where x _i = 0, 1, we define the weight w(x) of x by w(x) ?Σ _i=0 ?/sup> x _i . For any integer t with 0 < t < 2 k ?1, let S _t ?{(a, b) ??sup>2|a + b ?t (mod 2 k ?1), w(a) + w(b) < k, 0 ?a, b ?2 k ?2}. This paper gives explicit formulas for cardinality of S _t in the cases of w(t) ?3 and an upper bound for cardinality of S _t when w(t) = 4. From this one then concludes that a conjecture proposed by Tu and Deng in 2011 is true if w(t) ?4.