Cyclic codes over R=Fp+uFp+...+uk-1Fp with length psn

Authors:
Mu Han;Youpei Ye;Shixin Zhu;Chungen Xu;Bennian Dou
Affiliations:
School of Computer Science and Technology, Nanjing University of Science and Technology, Jiangsu 210094, China;School of Computer Science and Technology, Nanjing University of Science and Technology, Jiangsu 210094, China;School of Mathematics, Hefei University of Technology, Anhui 230009, China;School of Science, Nanjing University of Science and Technology, Jiangsu 210094, China;School of Science, Nanjing University of Science and Technology, Jiangsu 210094, China
Venue:
Information Sciences: an International Journal
Year:
2011

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Abstract

In this paper, all cyclic codes with length p^sn, (n prime to p) over the ring R=F"p+uF"p+...+u^k^-^1F"p are classified. It is first proved that Tor"j(C) is an ideal of S@?=F"p"^"m[@w]/, so that the structure of ideals over extension ring S"u"^"k(m,@w)=GR(u^k,m)[@w]/ is determined. Then, an isomorphism between R[X]/ and a direct sum @?"h"@?"IS"u"^"k(m"h,@w) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F"p+uF"p+...+u^k^-^1F"p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.