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Cyclic codes over GR(4m) which are also cyclic over Z4

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3 Author(s)
Junying Pei ; Sch. of Sci., Xidian Univ., Xi''an, China ; Jie Cui ; Sanyang Liu

Let GR(4m) be the Galois ring of characteristic 4 and cardinality 4m, and α_={α01,...,αm-1} be a basis of GR(4m) over Z4 when we regard GR(4m) as a free Z4-module of rank m. Define the map dα_ from GR(4m)[z]/(zn-1) into Z4[z]/(zmn-1) by dα_(a(z))=Σi=0m-1Σj=0n-1aijzmj+i where a(z)=Σj=0n-1ajzj and aji=0m-1aijαi, aij∈Z4. Then, for any linear code C of length n over GR(4m), its image dα_(C) is a Z4-linear code of length mn. In this article, for n and m being odd integers, it is determined all pairs (α_,C) such that dα_(C) is Z4-cyclic, where α_ is a basis of GR(4m) over Z4, and C is a cyclic code of length n over GR(4m).

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Information Theory, IEEE Transactions on  (Volume:49 ,  Issue: 3 )