Cyclic codes over \({\mathbb {F}}_{2^m}[u]/\langle u^k\rangle \) of oddly even length

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• 作者：Yonglin Cao ; Yuan Cao ; Fang-Wei Fu
• 刊名：Applicable Algebra in Engineering, Communication and Computing
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：27
• 期：4
• 页码：259-277
• 全文大小：520 KB
• 刊物类别：Computer Science
• 刊物主题：Symbolic and Algebraic Manipulation
  Computer Hardware
  Theory of Computation
  Artificial Intelligence and Robotics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0622
• 卷排序：27

文摘

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 and \(R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}\) (\(u^k=0\)) where \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring \(R[x]/\langle x^{2n}-1\rangle \). In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.KeywordsCyclic codeFinite chain ringNon-principal ideal ringDual codeSelf-dual code