Abstract
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.
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Garg, A., Dutt, S. (2017). On Rank and MDR Cyclic Codes of Length \(2^k\) Over \(Z_8\) . In: Gaur, D., Narayanaswamy, N. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2017. Lecture Notes in Computer Science(), vol 10156. Springer, Cham. https://doi.org/10.1007/978-3-319-53007-9_16
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DOI: https://doi.org/10.1007/978-3-319-53007-9_16
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