Repeated-root constacyclic codes of prime power lengths over finite chain rings

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• 作者：Hai Q. Dinh^a ; ^{hdinh@kent.edu} ; Hien D.T. Nguyen^b ; ^{nguyenthiduchiendhv@gmail.com} ; Songsak Sriboonchitta^c ; ^{songsakecon@gmail.com} ; Thang M. Vo^d ; ^{vomanhthang@vute.edu.vn}
• 关键词：primary ; 94B15 ; 94B05 ; secondary ; 11T71
• 刊名：Finite Fields and Their Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：43
• 期：Complete
• 页码：22-41
• 全文大小：436 K
• 卷排序：43

文摘

We study the algebraic structure of repeated-root λ  -constacyclic codes of prime power length psps over a finite commutative chain ring R   with maximal ideal 〈γ〉〈γ〉. It is shown that, for any unit λ of the chain ring R  , there always exists an element r∈Rr∈R such that λ−rpsλ−rps is not invertible, and furthermore, the ambient ring R[x]〈xps−λ〉 is a local ring with maximal ideal 〈x−r,γ〉〈x−r,γ〉. When there is a unit λ0λ0 such that λ=λ0ps, the nilpotency index of x−λ0x−λ0 in the ambient ring R[x]〈xps−λ〉 is established. When λ=λ0ps+γw, for some unit w of R  , it is shown that the ambient ring R[x]〈xps−λ〉 is a chain ring with maximal ideal 〈xps−λ0〉〈xps−λ0〉, which in turn provides structure and sizes of all λ-constacyclic codes and their duals. Among other things, situations when a linear code over R is both α- and β-constacyclic, for different units α, β, are discussed.