环Ｆ2+uＦ2上长为2e的循环码

近十多年来,有限环上的循环码一直是编码研究者所关心的热点问题,本文证明了R[x]/＜xn-1＞不是主理想环,其中R=F2 uF2,u2=0且n=2e.分3种情形讨论了环R[x]/＜xn-1＞中的非零理想,并给出了R上循环码的可以唯一确定的生成元的表达形式,同时给出了R上循环码的李距离的一个上界估计.     

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.     

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F2[u]/(u4)上的一类常循环码及其Gray象

该文定义了环R=F2+uF2+u2F2+u3F2到F24的一个新的Gray映射,其中u4 =0.证明了R上长为n的(1+u+u2 +u3)-循环码的Gray象是F2上长为4n的距离不变的线性循环码.进一步确定了R上奇长度的该常循环码的Gray象的生成多项式,并得到了一些最优的二元线性循环码.     

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+…+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.     

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＋ 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。     

Skew cyclic codes over ring F p +vF p

In this paper, we study skew cyclic codes over the ring F ＋ vFp, where p is a odd prime and v2 = 1. We give the generators of skew cyclic codes, with the consideration of the dual of skew cyclic codes.     

环F2+uF2上线性码及其对偶码的二元象

利用环F2＋uF2上线性码C的生成矩阵给出了码C的对偶码C^┴及其Gray象Ф（C）的生成矩阵，证明了环F2＋uF2上线性码及其对偶码的Gray象仍是对偶码。并由此给出了一个环F2＋uF2如上线性码为自对偶码的充要条件。     

有限非链环Fpm+vFpm上循环码的迹码和子环子码

给出了有限非链环F_p^m+vF_p^m的Galois扩环相关理论,明确了有限非链环F_p^m+vF_p^m上循环码的迹码和子环子码的生成元结构.     

Constacyclic codes of arbitrary lengths over ring Z_{p^m } + vZ_{p^m }

By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R = Zpm ＋ vZpm are studied, where v2 = 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）αp）-constacyclic codes over R are classified in terms of generator polynomial, where α is a unit of Zpm.     

Algebraic properties of BCH codes useful for decoding

In this paper theorems are presented which allow the simplified decoding of (n, k, δ) BCH codes in certain cases of practical interest. Such results are in a way implicit in the theory of BCH codes, but so far have not appeared explicitly in the literature. It is shown that any t₀ errors, 1 ? t₀ ? δ-1, can be detected by using any set of only t₀ consecutive coefficients of the syndrome polynomial. The correction of any t₀ errors, 1 ? t₀ ? [(δ-1)/2], can be performed by using any set of 2t₀ consecutive coefficients of the syndrome polynomial, where [x] means the integer part of x. Similar results are derived for punctured BCH codes. In this case sets of t₀ or 2t₀ consecutive coefficients, respectively, for detecting or correcting t₀ errors, are selected from the δ-1-p higher-order coefficients of the modified syndrome polynomial, where p is the number of digits punctured from a code word. These results hold true even when the punctured digits are not consecutive.     

非主理想环F_p+vF_p上线性码的MacWilliams恒等式

研究了环F-p+vF_p上线性码的结构,证明了互为对偶的线性码的Gray象仍是互为对偶的线性码.定义了环F_p+vF_p上码的Lee重量、Hamming重量和广义对称重量分布计数器的概念,利用域F_p上线性码和对偶码重量分布的关系及Gray映射的性质,给出了该环上线性码及其对偶码之间的各种重量分布的Macwilliam...     

Low rate convolutional and turbo codes based on non‐linear cyclic codes

Low rate convolutional and turbo codes that output non‐linear cyclic (NLC) codewords of length n = 2^m, m being a positive integer, are described. These codes have a very low coding rate, which makes them especially suitable for spread spectrum systems where they can be used for simultaneously achieving error correction and bandwidth expansion. Due to the cyclic properties and codeword length of the component codes, branch metrics can be efficiently computed using the fast Fourier transform (FFT), enabling simple implementation of the encoder and decoder. Among the possible NLC base codes, special attention is given to the Tomlinson, Cercas, Hughes (TCH) codes family due to their good autocorrelation properties. It is shown by simulation that the turbo codes schemes studied usually perform better than traditional turbo codes (in this paper the universal mobile telecommunications system (UMTS), rate 1/3 turbo code was used as a reference). This improvement is accomplished at the cost of bandwidth expansion. One of the advantages of the presented solutions over other low rate codes is their ability to improve the synchronization process at the receiver due to the good autocorrelation properties of the available NLC codes (especially TCH codes). A comparison of performance between the UMTS uplink connection and an equivalent system using the proposed codes for a multiuser scenario in a multipath fading channel is presented showing the possibility of capacity increase when using these codes. Copyright © 2006 John Wiley & Sons, Ltd.     

The Extended k-tree Algorithm

Consider the following problem: Given k=2 ^q random lists of n-bit vectors, L ₁,…,L _k , each of length m, find x ₁∈L ₁,…,x _k ∈L _k such that x ₁+⋅⋅⋅+x _k =0, where + is the XOR operation. This problem has applications in a number of areas, including cryptanalysis, coding theory, finding shortest lattice vectors, and learning theory. The so-called k-tree algorithm, due to Wagner, solves this problem in [(O)\tilde](2^q+n/(q+1))\tilde{O}(2^{q+n/(q+1)}) expected time provided the length m of the lists is large enough, specifically if m≥2 ^n/(q+1).