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A Class of Cyclic Codes Over ZPk+1

Author: LiLi
Tutor: DongXueDong
School: Liaoning Normal University
Course: Applied Mathematics
Keywords: Gray map Nechaev-Gray map Zpk+1- linear codes Zpk+1 - cyclic codes
CLC: O157.4
Type: Master's thesis
Year: 2008
Downloads: 20
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Abstract


Content: Cyclic Codes are a class of important error-correcting codes.It is known that the numbers of cyclic codes over finite rings are more than those of linear codes over finite fields with the same parameters such as length and minimum distance.Consequently cyclic codes over finite rings have better information rates.Therefore cyclic codes over finite rings have been extensively studied.So far,there has been much achievement about cyclic codes over the ring Z4. Structure and some properties of Z4-cyclic codes under the Gray map are given. Although the Gray map has been defined for codes over Zpk+1 and some properties of linear codes over Zpk+1 are given in some reference literatures,there has been little research on the structure of linear cyclic codes over Zpk+1 of length n,where n and p are coprime, via the Nechaev-Gray map.On the basis of known results, this paper discusses a class of cyclic codes obtained by using p-adic linear cyclic codes.Their structures of the Nechaev-Gray images are determined and the relation between p-adic linear cyclic codes and Zpk+1 -cyclic codes is given.Firstly,we study the p-adic linear cyclic codes of length pn,they are Nechaev-Gray images of Zp2-cyclic codes constructed by using p-adic linear cyclic codes of length n.Secondly, we discuss a family of cyclic codes obtained by using p-adic linear cyclic codes over Zpk+1.We determine their structures of the Nechaev-Gray images.By the method of inclusion we can conclude that p-adic cyclic codes of index pk-1 and of length pkn over Zp are Nechaev-Gray images of Zpk+1 -cyclic codes.Finally, the condition that cyclic codes are linear codes is given.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Combinatorics ( combinatorics ) > Coding theory ( on behalf of the digital theory )
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