This paper studies （1 ＋ u）-constacyelic codes over the ring F2 ＋ uF2 ＋ vF,2 ＋ uvF2. It is proved that the image of a （1 ＋ u）-constacyclic code of length n over F2 ＋ uF2 ＋ vF2 ＋uvF2 under a Gray map is a di...This paper studies （1 ＋ u）-constacyelic codes over the ring F2 ＋ uF2 ＋ vF,2 ＋ uvF2. It is proved that the image of a （1 ＋ u）-constacyclic code of length n over F2 ＋ uF2 ＋ vF2 ＋uvF2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined, Some optimal binary codes are obtained directly from （1 ＋ u）-constacyclic codes over F2 ＋ uF2 ＋ vF2 ＋ uvF2.展开更多
The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the ...The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the finite chain ring is proved to be a distance invariant quasi-cyclic code over the finite field. It is shown that every code over the finite field, which is the Gray image of a cyclic code over the finite chain ring, is equivalent to a quasi-cyclic code.展开更多
Abstract We investigate negacyclic codes over the Galois ring GR（2a,m） of length N = 2kn, where n is odd and k≥0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR（...Abstract We investigate negacyclic codes over the Galois ring GR（2a,m） of length N = 2kn, where n is odd and k≥0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR（2a, m）[u]/〈u2k ＋ 1〉. Then using a ring isomorphism we obtain the structure of negacyclic codes over GR（2a, m） of length N = 2kn （n odd） and explore the existence of self-dual negacyclic codes over GR（2a, m）. A bound for the homogeneous distance of such negacvclic codes is also given.展开更多
In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, ...In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li,P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.展开更多
Let t ≥ 2 be an integer, and let p1, ···, ptbe distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogo...Let t ≥ 2 be an integer, and let p1, ···, ptbe distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring Zp1p2···ptand the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over Zp1p2···pt.展开更多
By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and suffi...By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.展开更多
In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then...In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then define a Gray map between codes over the finite ring and a finite field. We prove that the Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field. We also prove that each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code.展开更多
Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,(1 — uv)-constacyclic codes over the local ring F_p + uF...Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,(1 — uv)-constacyclic codes over the local ring F_p + uF_p + vF_p + uvF_p are studied.It is proved that the image of a(1 — uv)-constacyclic code of length n over F_p + uF_p + vF_p + uvF_p under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p^3n over F_p.Several examples of optimal linear codes over F_p from(1 — uv)-constacyclic codes over F_p + uF_p + vF_p + uvF_p are given.展开更多
By the discussion of division in F2m[u]/〈u4〉,the minimal spanning set and the rank of a（1＋u＋u2）-constacyclic code with an arbitrary length N=2en over F2m[u]/〈u4〉 are determined based on the factorization of（x...By the discussion of division in F2m[u]/〈u4〉,the minimal spanning set and the rank of a（1＋u＋u2）-constacyclic code with an arbitrary length N=2en over F2m[u]/〈u4〉 are determined based on the factorization of（xn-1） over F2m.展开更多
We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v...We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.展开更多
In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized...In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes.展开更多
By constructing a Gray map, a class of constacyclic codes over ring R = R＋ vR is studied. Using cyclic codes and negacyclic codes of length p^s over ring R, the structure of （1 - 2v）-constacyclic codes and dual cod...By constructing a Gray map, a class of constacyclic codes over ring R = R＋ vR is studied. Using cyclic codes and negacyclic codes of length p^s over ring R, the structure of （1 - 2v）-constacyclic codes and dual codes of length p^s over ring R are given, the Gray images of （1 - 2v）-constacyclic codes in a particular case are also studied. It is shown that linear codes of length pS over ring R are （1 -2v）-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p^s over ring R.展开更多
In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic cod...In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4.展开更多
Abstract Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp ＋ uFp ...Abstract Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp ＋ uFp ＋ u^2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear （1 ＋ u ＋ u2）-constacyclic code over Fp ＋ uFp ＋ u^2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from （1 ＋ u ＋ u2）-constacyclic codes over F3 ＋ uF3 ＋ u^2F3 are given.展开更多
This paper consider Hexagonal-metric codes over certain class of finite fields. The Hexagonal metric as defined by Huber is a non-trivial metric over certain classes of finite fields. Hexagonal-metric codes are applie...This paper consider Hexagonal-metric codes over certain class of finite fields. The Hexagonal metric as defined by Huber is a non-trivial metric over certain classes of finite fields. Hexagonal-metric codes are applied in coded modulation scheme based on hexagonal-like signal constellations. Since the development of tight bounds for error correcting codes using new distance is a research problem, the purpose of this note is to generalize the Plotkin bound for linear codes over finite fields equipped with the Hexagonal metric. By means of a two-step method, the author presents a geometric method to construct finite signal constellations from quotient lattices associated to the rings of Eisenstein-Jacobi （E J） integers and their prime ideals and thus naturally label the constellation points by elements of a finite field. The Plotkin bound is derived from simple computing on the geometric figure of a finite field.展开更多
In this paper,we studied the depth spectrum and the depth distribution of constacyclic codes over the non-chain ring R=F_(p)+vF_(p)+v^(2)F_(p),where v^(3)=v.By decomposing the linear codes C over R into the linear cod...In this paper,we studied the depth spectrum and the depth distribution of constacyclic codes over the non-chain ring R=F_(p)+vF_(p)+v^(2)F_(p),where v^(3)=v.By decomposing the linear codes C over R into the linear codes over the finite field F_(p),three corresponding constacyclic codes C_(1),C_(2),C_(3) over F_(p)were obtained.Furthermore,considering the depth spectrum of constacyclic codes over the finite filed F_(p),and the relationship between constacyclic codes C_(1),C_(2),C_(3) and C,the depth spectrum and the depth distribution of constacyclic codes over R were discussed.展开更多
Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all...Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.展开更多
For any odd prime p such that p^m ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring Ra=Fpm[u]/〈u^a〉=Fp^m+uFp^m+…+u^a1Fp^m that have the form Λ=Λ0 + uΛ1 ++ u^a1 Λa1, where Λ0,Λ1,…,Λa1 ...For any odd prime p such that p^m ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring Ra=Fpm[u]/〈u^a〉=Fp^m+uFp^m+…+u^a1Fp^m that have the form Λ=Λ0 + uΛ1 ++ u^a1 Λa1, where Λ0,Λ1,…,Λa1 p^m,Λ0 ≠ 0,Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4p^s over a is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4p^s is expressed as a direct sum of a λ-constacyclic code and a λ-constacyclic code of length 2p^s. In the main case that the unit Λ is not a square, we prove that the polynomial x^4 λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where λ^p^s0=Λ0. From this, the ambient ring Ra[x]/<x^4p^sΛ>is proven to be a principal ideal ring, whose maximal ideals are <x^2 + 2ηx + 2η^2> and <x^2 2ηx + 2η^2>, where λ0 =4η^4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4p^s over a. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.展开更多
We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or ...We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R.Moreover,we discuss possible extension of our results in the more general setting ofδR-dual skew constacyclic codes over R,whereδR is an automorphism of R.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.60973125the Natural Science Foundation of Anhui Province under Grant No.1208085MA14the Fundamental Research Funds for the Central Universities under Grants Nos.2012HGXJ0040 and 2011HGBZ1298
文摘This paper studies （1 ＋ u）-constacyelic codes over the ring F2 ＋ uF2 ＋ vF,2 ＋ uvF2. It is proved that the image of a （1 ＋ u）-constacyclic code of length n over F2 ＋ uF2 ＋ vF2 ＋uvF2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined, Some optimal binary codes are obtained directly from （1 ＋ u）-constacyclic codes over F2 ＋ uF2 ＋ vF2 ＋ uvF2.
基金supported by the National Natural Science Foundation of China(60773002,60672119 and 60873144)the Program for New Century Excellent Talents in University,the Scientific Research Foundation for the Returned Overseas Chinese Scholars,the Hi-Tech Research and Development Program of China(2007AA01Z472)
文摘The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the finite chain ring is proved to be a distance invariant quasi-cyclic code over the finite field. It is shown that every code over the finite field, which is the Gray image of a cyclic code over the finite chain ring, is equivalent to a quasi-cyclic code.
基金supported by National Natural Science Foundation of China (Grant No. 60973125)College Doctoral Funds of China (Grant No. 20080359003)+1 种基金the Fundamental Research Funds for the Central Universities (Grant No. 2011HGXJ1079)the open research fund of National Mobile Communications Research Laboratory, Southeast University
文摘Abstract We investigate negacyclic codes over the Galois ring GR（2a,m） of length N = 2kn, where n is odd and k≥0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR（2a, m）[u]/〈u2k ＋ 1〉. Then using a ring isomorphism we obtain the structure of negacyclic codes over GR（2a, m） of length N = 2kn （n odd） and explore the existence of self-dual negacyclic codes over GR（2a, m）. A bound for the homogeneous distance of such negacvclic codes is also given.
基金supported by the National Natural Science Foundation of China(Nos.11171150,113711138,11531002)the Foundation of Science and the Technology on Information Assurance Laboratory(No.KJ-15-009)
文摘In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li,P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.
基金supported by the Project of Science and Technology Department of Sichuan Province(No.2016JY0134)
文摘Let t ≥ 2 be an integer, and let p1, ···, ptbe distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring Zp1p2···ptand the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over Zp1p2···pt.
基金Supported by the National Natural Science Foundation of China(No.61370089)
文摘By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.
基金supported by Anhui College Natural Science Research Project (KJ2013B221, 2012QRL156)Hefei Normal University General Research Project (2012kj10)Chuzhou University Research Project (2011kj002)
文摘In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then define a Gray map between codes over the finite ring and a finite field. We prove that the Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field. We also prove that each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code.
基金supported by the National Natural Science Foundation of China under Grant No.61370089the Natural Science Foundation of Anhui Province under Grant No.1208085MA14+2 种基金the Natural Science Fund of Education Department of Anhui province under Grant No.KJ2013Z276the Fundamental Research Fundsof Hefei University under Grant No.10KY01ZDthe Key construction discipline Funds of Hefei University under Grant No.2014XK08
文摘Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,(1 — uv)-constacyclic codes over the local ring F_p + uF_p + vF_p + uvF_p are studied.It is proved that the image of a(1 — uv)-constacyclic code of length n over F_p + uF_p + vF_p + uvF_p under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p^3n over F_p.Several examples of optimal linear codes over F_p from(1 — uv)-constacyclic codes over F_p + uF_p + vF_p + uvF_p are given.
基金Supported by the Natural Science Foundation of Anhui Province(KJ2015A308,KJ2016A307,1408085QF116)Anhui Province Colleges Outstanding Young Talents Program(gxyq ZD2016389,[2014]181)the Natural Science Project of Anhui Xinhua University(2014Zr009)
文摘By the discussion of division in F2m[u]/〈u4〉,the minimal spanning set and the rank of a（1＋u＋u2）-constacyclic code with an arbitrary length N=2en over F2m[u]/〈u4〉 are determined based on the factorization of（xn-1） over F2m.
文摘We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.
文摘In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes.
基金supported by the National Natural Science Foundation of China under Grant No.61370089
文摘By constructing a Gray map, a class of constacyclic codes over ring R = R＋ vR is studied. Using cyclic codes and negacyclic codes of length p^s over ring R, the structure of （1 - 2v）-constacyclic codes and dual codes of length p^s over ring R are given, the Gray images of （1 - 2v）-constacyclic codes in a particular case are also studied. It is shown that linear codes of length pS over ring R are （1 -2v）-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p^s over ring R.
文摘In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4.
基金supported by the National Natural Science Foundation of China under Grant No.61370089the Open Research Fund of National Mobile Communications Research Laboratory,Southeast University under Grant No.2014D04+3 种基金the Natural Science Fund of Education Department of Anhui province under Grant No.KJ2013Z276the Fundamental Research Funds of Hefei University under Grant No.10KY01ZDthe Key construction discipline Funds of Hefei University under Grant No.2014XK08the Natural Science Key Fund of Education Department of Anhui Province under Grant No.KJ2015A226
文摘Abstract Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp ＋ uFp ＋ u^2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear （1 ＋ u ＋ u2）-constacyclic code over Fp ＋ uFp ＋ u^2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from （1 ＋ u ＋ u2）-constacyclic codes over F3 ＋ uF3 ＋ u^2F3 are given.
基金supported by 973 project under Grant No.2007CB807901the Fundamental Research Funds for the Central Universities under Grant Nos.YWFF-10-02-072 and YWF-10-01-A28
文摘This paper consider Hexagonal-metric codes over certain class of finite fields. The Hexagonal metric as defined by Huber is a non-trivial metric over certain classes of finite fields. Hexagonal-metric codes are applied in coded modulation scheme based on hexagonal-like signal constellations. Since the development of tight bounds for error correcting codes using new distance is a research problem, the purpose of this note is to generalize the Plotkin bound for linear codes over finite fields equipped with the Hexagonal metric. By means of a two-step method, the author presents a geometric method to construct finite signal constellations from quotient lattices associated to the rings of Eisenstein-Jacobi （E J） integers and their prime ideals and thus naturally label the constellation points by elements of a finite field. The Plotkin bound is derived from simple computing on the geometric figure of a finite field.
基金Supported by the Open Research Fund of Key Laboratory of Intelligent Computing and Signal Processing,Ministry of Education,Anhui University.
文摘In this paper,we studied the depth spectrum and the depth distribution of constacyclic codes over the non-chain ring R=F_(p)+vF_(p)+v^(2)F_(p),where v^(3)=v.By decomposing the linear codes C over R into the linear codes over the finite field F_(p),three corresponding constacyclic codes C_(1),C_(2),C_(3) over F_(p)were obtained.Furthermore,considering the depth spectrum of constacyclic codes over the finite filed F_(p),and the relationship between constacyclic codes C_(1),C_(2),C_(3) and C,the depth spectrum and the depth distribution of constacyclic codes over R were discussed.
基金Supported by Research Funds of Hubei Province(D20144401,Q20174503)。
文摘Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.
文摘For any odd prime p such that p^m ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring Ra=Fpm[u]/〈u^a〉=Fp^m+uFp^m+…+u^a1Fp^m that have the form Λ=Λ0 + uΛ1 ++ u^a1 Λa1, where Λ0,Λ1,…,Λa1 p^m,Λ0 ≠ 0,Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4p^s over a is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4p^s is expressed as a direct sum of a λ-constacyclic code and a λ-constacyclic code of length 2p^s. In the main case that the unit Λ is not a square, we prove that the polynomial x^4 λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where λ^p^s0=Λ0. From this, the ambient ring Ra[x]/<x^4p^sΛ>is proven to be a principal ideal ring, whose maximal ideals are <x^2 + 2ηx + 2η^2> and <x^2 2ηx + 2η^2>, where λ0 =4η^4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4p^s over a. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.
基金This work was supported by the Ministry of Education and Training of Vietnam(Thai Nguyen University)under Grant No.B2019-TNA-02.
文摘We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R.Moreover,we discuss possible extension of our results in the more general setting ofδR-dual skew constacyclic codes over R,whereδR is an automorphism of R.
