环F_(p~m)+uF_(p~m)+…+u~(k-1)F_(p~m)上 (1+u)-常循环码的齐次距离分布
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摘要
该文研究了环rK=F(pm)+uF(pm)+…+u(k1)F(pm)上任意长的(1+u)-常循环码的齐次距离分布。首先,介绍了环k R上给定长度的(1+u)-常循环码的挠码。然后利用挠码得到环k R上任意长度的(1+u)-常循环码的齐次距离的界,并给出了Rk上某些(1+u)-常循环码的齐次距离的准确值。
In this paper, the distribution of homogeneous distance of(1+u)-constacyclic codes over the ring Rk= F(pm)+uF(pm)+…+u(k-1)F(pm) of arbitrary lengths is studied. Firstly, the torsion codes of a(1+u)-constacyclic code over k R for a given length are introduced. Then, by using the torsion codes, a bound for the homogeneous distance of(1+u)-constacyclic codes over Rk of any length is given. The exact homogeneous distance of some(1+u)-constacyclic codes over Rk is also obtained.
In this paper, the distribution of homogeneous distance of(1+u)-constacyclic codes over the ring Rk= F(pm)+uF(pm)+…+u(k-1)F(pm) of arbitrary lengths is studied. Firstly, the torsion codes of a(1+u)-constacyclic code over k R for a given length are introduced. Then, by using the torsion codes, a bound for the homogeneous distance of(1+u)-constacyclic codes over Rk of any length is given. The exact homogeneous distance of some(1+u)-constacyclic codes over Rk is also obtained.
引文
[1]Dinh H Q.Negacyclic codes of length 2sover Galois rings[J].IEEE Transactions on Information Theory,2005,51(4):4252-4262.
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[6]彭培让,郑喜英,孔波.环4Z上长为2s的循环码的齐次距离[J].河南大学学报(自然科学版),2012,42(2):121-124. Peng P R,Zheng X Y,and Kong B.Homogeneous distance of cyclic codes of length 2son ring4Z[J].Journal of Henan University(Natural Science),2012,42(2):121-124.
[7]邓林,朱士信,韩江洪.环2 2F+uF上长为2s的(1+u)-常循环码的距离分布[J].中国科技大学学报,2008,38(10):1810-1814.Deng L,Zhu S X,and Han J H.The distribution of distances of(1+u)-constacyclic codes of length 2sover2 2F+uF[J].Journal of University of Science and Technology of China,2008,38(10):1810-1814.
[8]施敏加,杨善林,朱士信.环12 2 2kF uF u F-+++上长为2s的(1+u)-常循环码的距离分布[J].电子与信息学报,2010,32(1):112-116.Shi M J,Yang S L,and Zhu S X.The distributions of distances of(1+u)-constacyclic codes of length 2sover12 2 2kF uF u F-+++[J].Journal of Electronics&Information Technology,2010,32(1):112-116.
[9]施敏加,杨善林,朱士信.环2 2F+uF上长度为2s的循环码的距离[J].电子学报,2011,39(1):29-34.Shi M J,Yang S L,and Zhu S X.On minimum distances of cyclic codes of length 2sover2 2F+uF[J].Acta Electronic Sinica,2011,39(1):29-34.
[10]Zhu S X and Kai X S.Negacyclic codes over Galois rings of characteristic 2a[J].Science China Mathematics,2012,55(4):869-879.
[11]刘晓娟,朱士信.环1+++m m m k p p p F uF u F-上的长为sp的(1+lu)常循环码的距离分布[J].中国科学技术大学学报,2012,42(11):931-935.Liu X J and Zhu S X.The distributions of distances of(1+lu)-constacyclic codes of lengthsp over+m mp p F uF1++m k p u F-[J].Journal of University of Science and Technology of China,2012,42(11):931-935.
[12]Dinh H Q and Nguyen H D.On some classes of constacyclic codes over polynomial residue rings[J].Mathematics of Communications,2012,6(2):175-191.
[13]Han M,Ye Y P,Zhu S X,et al..Cyclic codes over k1p p p R F uF u F-=+++with lengthsp n[J].Information Science,2011,181(4):926-934.
[14]Sǎlǎgean A.Repeated-root cyclic and negacyclic codes over a finite chain ring[J].Discrete Applied Mathematics,2005,154(2):413-419.
[15]Castagnoli G,Massey J L,Schoeller P A,et al..On repeated-root cyclic codes[J].IEEE Transactions on Information Theory,1991,37(2):337-342.
[2]Dinh H Q.Complete distance of all negacyclic codes of length2sover2aZ[J].IEEE Transactions on Information Theory,2007,53(1):147-161.
[3]Norton G H and Sǎlǎgean A.On the Hamming distance of linear codes over a finite chain ring[J].IEEE Transactions on Information Theory,2000,46(3):1060-1067.
[4]Zhu S X and Kai X S.The Hamming distances of negacyclic codes of length 2sover GR(2,)am[J].Journal of System Science and Complexity,2008,21(1):60-66.
[5]Kai X S and Zhu S X.On the distances of cyclic codes of length 2sover4Z[J].Discrete Mathematics,2010,310(1):12-20.
[6]彭培让,郑喜英,孔波.环4Z上长为2s的循环码的齐次距离[J].河南大学学报(自然科学版),2012,42(2):121-124. Peng P R,Zheng X Y,and Kong B.Homogeneous distance of cyclic codes of length 2son ring4Z[J].Journal of Henan University(Natural Science),2012,42(2):121-124.
[7]邓林,朱士信,韩江洪.环2 2F+uF上长为2s的(1+u)-常循环码的距离分布[J].中国科技大学学报,2008,38(10):1810-1814.Deng L,Zhu S X,and Han J H.The distribution of distances of(1+u)-constacyclic codes of length 2sover2 2F+uF[J].Journal of University of Science and Technology of China,2008,38(10):1810-1814.
[8]施敏加,杨善林,朱士信.环12 2 2kF uF u F-+++上长为2s的(1+u)-常循环码的距离分布[J].电子与信息学报,2010,32(1):112-116.Shi M J,Yang S L,and Zhu S X.The distributions of distances of(1+u)-constacyclic codes of length 2sover12 2 2kF uF u F-+++[J].Journal of Electronics&Information Technology,2010,32(1):112-116.
[9]施敏加,杨善林,朱士信.环2 2F+uF上长度为2s的循环码的距离[J].电子学报,2011,39(1):29-34.Shi M J,Yang S L,and Zhu S X.On minimum distances of cyclic codes of length 2sover2 2F+uF[J].Acta Electronic Sinica,2011,39(1):29-34.
[10]Zhu S X and Kai X S.Negacyclic codes over Galois rings of characteristic 2a[J].Science China Mathematics,2012,55(4):869-879.
[11]刘晓娟,朱士信.环1+++m m m k p p p F uF u F-上的长为sp的(1+lu)常循环码的距离分布[J].中国科学技术大学学报,2012,42(11):931-935.Liu X J and Zhu S X.The distributions of distances of(1+lu)-constacyclic codes of lengthsp over+m mp p F uF1++m k p u F-[J].Journal of University of Science and Technology of China,2012,42(11):931-935.
[12]Dinh H Q and Nguyen H D.On some classes of constacyclic codes over polynomial residue rings[J].Mathematics of Communications,2012,6(2):175-191.
[13]Han M,Ye Y P,Zhu S X,et al..Cyclic codes over k1p p p R F uF u F-=+++with lengthsp n[J].Information Science,2011,181(4):926-934.
[14]Sǎlǎgean A.Repeated-root cyclic and negacyclic codes over a finite chain ring[J].Discrete Applied Mathematics,2005,154(2):413-419.
[15]Castagnoli G,Massey J L,Schoeller P A,et al..On repeated-root cyclic codes[J].IEEE Transactions on Information Theory,1991,37(2):337-342.