环R=F_3+vF_3(v~2=1)上线性码的深度谱
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摘要
设q为素数的方幂,F_q为q元有限域.本文通过将F_q上向量的深度概念推广到环R=F_3+vF_3(v~2=1)上,给出R上线性码码字深度的递归算法,进而利用环R上线性码的生成矩阵及环R到F_3的两个加群同态,给出环R上任意长度的线性码深度谱的上下界.并由此推出环R_1=F_P+vF_P(v~2=1)上任意长度的非零线性码深度谱的上下界,其中p为奇质数.
Let q be a power of a prime and Fq be the q elements finite field. By generalizing the concept for the depth of vectors from Fq to the ring R = F_3 + vF3(v2 = 1),a recursive algorithm for the depth of the vector over R is obtained. Furthermore, by defining two homomorphisms from the Abelian group R to F_3, the upper bound and lower bound for the depth spectra of linear codes with arbitrary code length over R are obtained based on the generator matrices. And thus the upper bound and lower bound for the depth spectra of linear codes with arbitrary code length over R_1= F_p +vF_p(v~2 = 1) can be derived, where p is an odd prime.
Let q be a power of a prime and Fq be the q elements finite field. By generalizing the concept for the depth of vectors from Fq to the ring R = F_3 + vF3(v2 = 1),a recursive algorithm for the depth of the vector over R is obtained. Furthermore, by defining two homomorphisms from the Abelian group R to F_3, the upper bound and lower bound for the depth spectra of linear codes with arbitrary code length over R are obtained based on the generator matrices. And thus the upper bound and lower bound for the depth spectra of linear codes with arbitrary code length over R_1= F_p +vF_p(v~2 = 1) can be derived, where p is an odd prime.
引文
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[3]Luo,Y.,Fu,F.W.and Wei,V.K.W.,On the depth distribution of linear codes,IEEE Trans.Inform.Theory,2000,46(6):2197-2203.
[4]Mitchell,J.C.,On integer valued rational polynomials and depth distribution of binary code,IEEE Trans.Inform.Theory,1998,44(7):3146-3150.
[5]Pu,K.L.and Liao,Q.Y.,The depth distribution and spectrum of linear codes over the ring R=F_q+uF_q(u~2=0),Adv.Math.(China),2014,43(1):57-63(in Chinese).
[6]Shi,M.J.and Yang,S.L.,MacWilliams identities of linear codes over non-principal ideal ring F_q+vF_q,Acta Electronica Sinica,2011,39(10):2449-2453(in Chinese).
[7]Tang,G.,On the depth spectra of linear codes on ring R=F_2+vF_2,J.Math.(Wuhan),2012,32(1):186-190(in Chinese).
[8]Yang,S.L.,Zhu,S.X.and Tong,H.X.,Two recursive algorithms for computing the depth of a codeword on finite ring Z4,J.Univ.Sci.Technol.China,2004,34(6):655-600(in Chinese).
[9]Yao,T.,Shi,M.J.and Chen,Y.,The depth spectra of linear codes over F_2+uF_2 u~2F_2,IEICE Trans.Fundamentals,2016,E99-A(1):429-432.
[10]Yu,H.F.and Zhu,S.X.,Depth distribution of linear codes over ring R=F_2+uF_2,J.Univ.Sci.Technol.China,2008,38(2):141-144(in Chinese).
[11]Zhu,S.X.,Yang,S.L.and Tong,H.X.,On the depth spectrums of linear cyclic codes on ring Z_4,J.Electron.Inform.Technol.,2005,27(10):1597-1599(in Chinese).
[2]Etzion,T.,The depth distribution一a characterization for linear codes,IEEE Trans.Inform.Theory,1997,43(4):1361-1363.
[3]Luo,Y.,Fu,F.W.and Wei,V.K.W.,On the depth distribution of linear codes,IEEE Trans.Inform.Theory,2000,46(6):2197-2203.
[4]Mitchell,J.C.,On integer valued rational polynomials and depth distribution of binary code,IEEE Trans.Inform.Theory,1998,44(7):3146-3150.
[5]Pu,K.L.and Liao,Q.Y.,The depth distribution and spectrum of linear codes over the ring R=F_q+uF_q(u~2=0),Adv.Math.(China),2014,43(1):57-63(in Chinese).
[6]Shi,M.J.and Yang,S.L.,MacWilliams identities of linear codes over non-principal ideal ring F_q+vF_q,Acta Electronica Sinica,2011,39(10):2449-2453(in Chinese).
[7]Tang,G.,On the depth spectra of linear codes on ring R=F_2+vF_2,J.Math.(Wuhan),2012,32(1):186-190(in Chinese).
[8]Yang,S.L.,Zhu,S.X.and Tong,H.X.,Two recursive algorithms for computing the depth of a codeword on finite ring Z4,J.Univ.Sci.Technol.China,2004,34(6):655-600(in Chinese).
[9]Yao,T.,Shi,M.J.and Chen,Y.,The depth spectra of linear codes over F_2+uF_2 u~2F_2,IEICE Trans.Fundamentals,2016,E99-A(1):429-432.
[10]Yu,H.F.and Zhu,S.X.,Depth distribution of linear codes over ring R=F_2+uF_2,J.Univ.Sci.Technol.China,2008,38(2):141-144(in Chinese).
[11]Zhu,S.X.,Yang,S.L.and Tong,H.X.,On the depth spectrums of linear cyclic codes on ring Z_4,J.Electron.Inform.Technol.,2005,27(10):1597-1599(in Chinese).