On Rank and MDR Cyclic Codes of Length \(2^k\) Over \(Z_8\)

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• 关键词：Cyclic codes ; Minimal degree polynomial ; Rank ; MDR codes
• 刊名：Lecture Notes in Computer Science
• 出版年：2017
• 出版时间：2017
• 年：2017
• 卷：10156
• 期：1
• 页码：177-186
• 丛书名：Algorithms and Discrete Applied Mathematics
• ISBN：978-3-319-53007-9
• 卷排序：10156

文摘

In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length \(n = 2^k\) (where k is a natural number) over \(Z_8\) is obtained. This set of generators is used to find the rank of the cyclic code C. It is proved that the rank of a cyclic code C of length \(n=2^k\) over \(Z_8\) is equal to \(n-v\), where v is the degree of a minimal degree polynomial in C. Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length \(n=2^k\) over \(Z_8\) is given. An example of the best codes over \(Z_8\) of length 4 having largest minimum Hamming, Lee and Euclidean distances among all codes of the same rank is also given.