On the lifted Zetterberg code

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• 作者：Adel Alahmadi ; Hussain Alhazmi ; Tor Helleseth…
• 刊名：Designs, Codes and Cryptography
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：80
• 期：3
• 页码：561-576
• 全文大小：1,222 KB
• 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
• 卷排序：80

文摘

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial \(g(x)=(x+1)p(x)\), where p(x) is the minimum polynomial over GF(2) of an element of order \(2^m+1\) in \(GF(2^{2m})\) and m is even. This even binary code has parameters \([2^m+1,2^m-2m, 6]\). The quaternary code obtained by lifting the code to the alphabet \({\mathbb {Z}}_4=\{0,1,2,3\}\) is shown to have parameters \([2^m+1,2^m-2m, d_L ]\), where \(d_L \ge 8\) denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters \((2^{m+1}+2,2^k,d_H)\), where \(d_H \ge 8\) denotes the minimum Hamming weight and \(k=2^{m+1}-4m\). For \(m=6\) these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these \({\mathbb {Z}}_4\)-codes for all even m. This appears to be the first infinite family of \({\mathbb {Z}}_4\)-codes of length \(n=2^m+1\) with \(d_L \ge 8\) having an algebraic decoding algorithm.KeywordsZetterberg codeCyclic codesCodes over \({\mathbb {Z}}_4\)