Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

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• 作者：Maosheng Xiong ; Nian Li ; Zhengchun Zhou ; Cunsheng Ding
• 关键词：Cyclic codes ; Weight distribution ; Vandermonde matrix ; Niho exponent ; 11T71 ; 94B15 ; 11L03
• 刊名：Designs, Codes and Cryptography
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：78
• 期：3
• 页码：713-730
• 全文大小：514 KB
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• 作者单位：Maosheng Xiong (1)
  Nian Li (1)
  Zhengchun Zhou (2)
  Cunsheng Ding (3)
  
  1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
  2. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China
  3. Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279–3286, 2010; IEEE Trans Inf Theory 60:3903–3912, 2014), we study two families of cyclic codes over \({\mathbb F}_p\) with arbitrary number of zeroes of generalized Niho type, more precisely \({\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}\) (for \(p=2\)) of \(t+1\) zeroes, and \({\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}\) (for any prime \(p\)) of \(t\) zeroes for any \(t\). We find that the first family has at most \((2t+1)\) non-zero weights, and the second has at most \(2t\) non-zero weights. Their weight distribution are also determined in the paper. Keywords Cyclic codes Weight distribution Vandermonde matrix Niho exponent