MDS codes over finite principal ideal rings

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• 作者：Steven T. Dougherty ; Jon-Lark Kim and Hamid Kulosman
• 关键词：Chain ring ; Galois ring ; MDS code ; Principal ideal ring
• 刊名：Designs, Codes and Cryptography
• 出版年：2009
• 出版时间：January, 2009
• 年：2009
• 卷：50
• 期：1
• 页码：77-92
• 全文大小：238.3 KB

文摘

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including \mathbbZ_pe{\mathbb{Z}_{p^e}}). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.