MacWilliams Extension Theorem for MDS codes over a vector space alphabet

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• 作者：Serhii Dyshko
• 关键词：MacWilliams Extension Theorem ; Code over a vector space alphabet ; Code isometry ; MDS code ; Near ; MDS code ; Monomial map ; Partition of a vector space
• 刊名：Designs, Codes and Cryptography
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：82
• 期：1-2
• 页码：57-67
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits;
• 出版者：Springer US
• ISSN：1573-7586
• 卷排序：82

文摘

The MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a monomial map. Unlike the linear codes, in general, nonlinear codes do not have the extension property. In our previous work, in the context of a vector space alphabet, the minimum code length, for which there exists an unextendable code isometry, was determined. In this paper an analogue of the extension theorem for MDS codes is proved. It is shown that for almost all, except 2-dimensional, linear MDS codes over a vector space alphabet the extension property holds. For the case of 2-dimensional MDS codes an improvement of our general result is presented. There are also observed extension properties of near-MDS codes. As an auxiliary result, a new bound on the minimum size of multi-fold partitions of a vector space is obtained.