Linear codes using skew polynomials with automorphisms and derivations

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• 作者：D. Boucher (1)
  F. Ulmer (1)
  
• 关键词：Error ; correcting codes ; Decoding ; Finite fields ; Skew polynomial rings ; 12Y05 ; 16Z05 ; 94B05
• 刊名：Designs, Codes and Cryptography
• 出版年：2014
• 出版时间：March 2014
• 年：2014
• 卷：70
• 期：3
• 页码：405-431
• 全文大小：354 KB
• 作者单位：D. Boucher (1)
  F. Ulmer (1)
  
  1. IRMAR (UMR 6625), Universit茅 de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France
  
• ISSN：1573-7586

文摘

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.