环Fpm+uFpm上长为pk的循环码计数

环R=Fpm+uFpm上长为pk的循环码可看作R[x]/＜xpk-1＞上的理想.该文通过对R[x]/＜xpk-1＞上理想的研究,得到了环Fpm+uFpm上长为的循环码的唯一表示方法和计数,并给出了该环上长为pk的循环自对偶码的结构和计数.     

环 F2＋ uF2＋ u2 F2上的常循环码

常循环码是一类重要的纠错码,本文基于（xn －1）在 F2[x]上的分解,探讨了环 R＝ F2＋ uF2＋ u2 F2上任意长度的（1＋λu）常循环码的极小生成元集（λ为R上的单位）。通过分析该环上循环码和常循环码的置换等价性,得到了该环上码长为奇数及码长 N≡2（mod 4）时（1＋ u2）常循环码的生成多项式和极小生成元集。     

环Fq+uFq+...+uh-1Fq上任意长度的(u?-1)-常循环码

该文利用环同态理论,给出了环k 1 q q q R F uF u F =++L+-上任意长度N 的所有(ul -1)-常循环码的生成元, l 是R 的可逆元.证明了[]/1 N R x < x +-ul >是主理想环.给出了环R上任意长度N 的(ul -1)-常循环码的计数.确定了环R上任意长度N 的(ul -1)-常循环码的最高阶挠码的生成多项式,由此给出了环R上长度 s p 的所有(ul -1)-常循环码的汉明距离.     

环F2+uF2上线性码的结构特性

该文研究了环F2 uF2上线性码的结构特性,讨论了环F2 uF2上线性码及其剩余码、挠码和商码之间的关系,通过这些关系.给出了线性码(特别是循环码)的深度分布与深度谱.     

环 F_2+uF_2+…+u~(κ-1)F_2上长为2~s的(1+u)-常循环码的距离分布

研究码字的距离分布是编码理论的一个重要研究方向。该文定义了环R=F2+uF2++uk-1F2上的Homogeneous重量,研究了环R上长为2s的(1+u)-常循环码的Hamming距离和Homogeneous距离。使用了有限环和域的理论,给出了环R上长为2s的(1+u)-常循环码和循环自对偶码的结构和码字个数。并利用该常循环码的结构,确定了环R上长为2s的(1+u)-常循环码的Hamming距离和Homogeneous距离分布。     

环Fq+uFq++uk-1Fq上一类重根常循环码

记R=Fq+uFq++uk-1Fq,G=R[x]/,且是R中可逆元。定义了从Gn到Rtn的新的Gray映射,证明了J是G上长为n的线性的x-常循环码当且仅当(J)是R上长为tn的线性的-常循环码。使用有限环理论,获得了环R上长为pe的所有的(u-1)-常循环码的结构及其码字个数。特别地,获得了环F2m+uF上长为2e的(u-1)-常循环码的对偶码的结构及其码字个数。推广了环Z2a根负循环码的若干结果。     

环F2+uF2+…+uk-1F2上常循环自对偶码

最近,剩余类环上的常循环码及常循环自对偶码引起了编码学者的极大关注.本文首先利用一些相关的线性码,建立了一类特殊有限链环上长为N的常循环自对偶码的一般理论,利用其结果给出了该环上长为N的(1+uλ)-常循环自对偶码存在的充分条件,得到了该环上长为N的一些常循环自对偶码,并给出了其生成多项式.     

Z4×(F2+uF2)上的一类循环码

定义了Z₄×（F₂+uF₂）上的循环码,明确了一类循环码的生成元结构,给出了该类循环码的极小生成元集.利用Gray映射,构造了一些二元非线性码.     

环F2+uF2上长度为2e的循环码的距离

确定码字的Hamming距离和Lee距离是解码的关键.本文对环F₂+uF₂上长度为2^e的循环码的结构进行了分类.确定了环F₂+uF₂上某些长度为2^e的循环码的Hamming距离和Lee距离.给出了环F₂+uF₂上长度为2^e的其它循环码的Hamming距离的上界及Lee距离的上界和下界.     

环Z2m上一类常循环码的挠码及其应用

该文研究了环Z_2^m上任意长的（1+2λ）-常循环码的挠码及其应用.首先,给出环Z_2^m上（1+2λ）-常循环码的挠码.然后,利用挠码得到环Z_2^m上某些（1+2λ）-常循环码的齐次距离分布.同时,利用挠码证明了环Z_2^m上（2^m－1－1）-常循环自对偶码都是类型I码,并利用这类码构造了极优的类型I码.     

Optimal quinary cyclic codes with minimum distance four

Cyclic codes are an extremely important subclass of linear codes.They are widely used in the communication systems and data storage systems because they have efficient encoding and decoding algorithm.Until now,how to construct the optimal ternary cyclic codes has received a lot of attention and much progress has been made.However,there is less research about the optimal quinary cyclic codes.Firstly,an efficient method to determine if cyclic codes C_(1,e,t)were optimal codes was obtained.Secondly,based on the proposed method,when the equation e=5_k+1 or e=5_m?2hold,the theorem that the cyclic codes C_(1,e,t)were optimal quinary cyclic codes was proved.In addition,perfect nonlinear monomials were used to construct optimal quinary cyclic codes with parameters[5_m?1,5_m?2m?2,4]optimal quinary cyclic codes over $F_{5^m}$.     

One and Two-Weight Z2R2 Additive Codes

This paper is devoted to the construction of one and two-weight Z2R2 additive codes, where R2 =F2[v]/. It is a generalization towards another direction of Z2Z4 codes (S.T. Dougherty, H.W. Liu and L. Yu,"One weight Z2Z4 additive codes", Applicable Algebra in Engineering, Communication and Computing, Vol.27, No.2, pp.123–138, 2016). A MacWilliams identity which connects the weight enumerator of an additive code over Z2 R2 and its dual is established. Several construction methods of one-weight and two-weight additive codes over Z2 R2 are presented. Several examples are presented to illustrate our main results and some open problems are also proposed.     

Algebraic lower bounds on the free distance of convolutional codes

A new module structure for convolutional codes is introduced and used to establish further links with quasi-cyclic and cyclic codes. The set of finite weight codewords of an (n,k) convolutional code over F_q is shown to be isomorphic to an F_q[x]-submodule of F_q ⁿ[x], where F_q ⁿ[x] is the ring of polynomials in indeterminate x over F_q ⁿ, an extension field of F_q. Such a module can then be associated with a quasi-cyclic code of index n and block length nL viewed as an F_q[x]-submodule of F_q ⁿ[x]/langx^L-1rang, for any positive integer L. Using this new module approach algebraic lower bounds on the free distance of a convolutional code are derived which can be read directly from the choice of polynomial generators. Links between convolutional codes and cyclic codes over the field extension F_q ⁿ are also developed and Bose-Chaudhuri-Hocquenghem (BCH)-type results are easily established in this setting. Techniques to find the optimal choice of the parameter L are outlined     

有限域上最优码的一种新的构造方法

基于域F_p^m上一类特殊的矩阵,定义了环R(_p^m,k)=F_p^m[u]/k>到F_p^p_mj的一个新的Gray映射,其中u^k=0、p为素数、j为正整数且p^j－1+1≤k≤p^j.得到了环R(_p^m,k)上码长为任意长度N的(1+u)常循环码的Gray象是F_p^m上长为p^jN的保距线性循环码,并给出了Gray象的生成多项式,构造了F₃,F₅和F₇上的一些最优线性循环码.     

环F2+uF2+vF2+uvF2上(1+uv)-循环码

该文定义了有限非链环R=F2+uF2+vF2+uvF2上(1+uv)-循环码的相关概念,讨论了其与该环上循环码的关系,证明了此环上(1+uv)-循环码在关于齐次重量的等距Gray映射hom下的二元象是一个长为8n的4-准循环码, 并由此映射得到了一些好的二元线性准循环码。     

环F2+uF2+…+uk-1F2上长为2S的(1+u)-常循环码的距离分布

研究码字的距离分布是编码理论的一个重要研究方向。该文定义了环R=F₂+uF₂+…+u^k-1F₂上的Homogeneous重量,研究了环R上长为2^S的(1+u)-常循环码的Hamming距离和Homogeneous距离。使用了有限环和域的理论,给出了环R上长为2^S的(1+u)-常循环码和循环自对偶码的结构和码字个数。并利用该常循环码的结构,确定了环R上长为2^S的(1+u)-常循环码的Hamming距离和Homogeneous距离分布。     

On the [24, 12, 10] quaternary code and binary codes with anautomorphism having two cycles

A general decomposition theorem is given for codes over finite fields which have an automorphism of a given type. Such codes can be decomposed as direct sums of subcodes which may be viewed as shorter length codes over extension fields. If such a code is self-dual, sometimes the subcodes are also. This decomposition is applied to prove that the self-dual [24, 12, 10] quaternary code has no automorphism of order 3. This decomposition is also applied to count the number of equivalent [2r, r] and [2r+2r+1] self-dual binary codes with an automorphism of prime order r     

Optimal covering polynomial sets correcting three errors for binarycyclic codes

The covering polynomial method is a generalization of error-trapping decoding and is a simple and effective way to decode cyclic codes. For cyclic codes of rate R<2/τ, covering polynomials of a single term suffice to correct up to τ errors, and minimal sets of covering polynomials are known for various such codes. In this article, the case of τ=3 and of binary cyclic codes of rate R⩾2/3 is investigated. Specifically, a closed-form specification is given for minimal covering polynomial sets for codes of rate 2/3⩽R<11/15 for all sufficiently large code length n; the resulting number of covering polynomials is, if R=2/3+ρ with ρ>0, equal to nρ+2V√nρ+(1/2) log_φ(n/ρ)+O(1), where φ=(1+√5)/2. For all codes correcting up to three errors, the number of covering polynomials is at least nρ+2√nρ+O(log n); covering polynomial sets achieving this bound (and thus within O(log n) of the minimum) are presented in closed-form specifications for rates in the range 11/15⩽R<3/4