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.     

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+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。     

环F4+uF4上线性码及其对偶码的Gray象

研究了环F4+uF4与域F4上的线性码,利用环F4+uF4上码C的Gray重量wG,Gray距离d G和(F4+uF4)n到F4 2n的Gray映射φ,证明了环F4+uF4上线性码C及其对偶码的Gray像φ(C)为F4上的线性码和对偶且dH G(φ(C))dG(C)。同时,给出了F4+uF4上循环码C的Gray像φ(C)为F4上的2-拟循环码。     

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.     

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.     

环Z4上自对偶码的构造

利用有限环Z₄+vZ₄（其中v²=1）上自对偶码,给出了一种构造Z₄上自对偶码的方法.引入了（Z₄+vZ₄）ⁿ到Z₄²ⁿ的保距Gray映射,给出了Z₄+vZ₄上自对偶码的性质,证明了Z₄+vZ₄上长为n的自对偶码的Gray像是Z₄上长为2n的自对偶码,由此构造了Z₄上一些极优的类型I与类型Ⅱ自对偶码.     

Negacyclic codes of length 2/sup s/ over galois rings

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet Zopf₄, and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2^s over Galois rings. The structure of negacyclic codes of length 2^s over the Galois rings GR(2^a,m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2^a,m) in general, and over Zopf₂ ^a in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over Zopf₂ ^a of length 4,8, and 16 are given. The weight distributions of such negacyclic codes are also discussed     

A new construction of 16-QAM Golay complementary sequences

We present a new construction of 16-QAM Golay sequences of length n = 2/sup m/. The number of constructed sequences is (14 + 12m)(m!/2)4/sup m+1/. When employed as a code in an orthogonal frequency-division multiplexing (OFDM) system; this set of sequences has a peak-to-mean envelope power ratio (PMEPR) of 3.6. By considering two specific subsets of these sequences, we obtain new codes with PMEPR bounds of 2.0 and 2.8 and respective code sizes of (2 + 2m)(m!/2)4/sup m+1/ and (4 + 4m)(m!/2)4/sup m+1/. These are larger than previously known codes for the same PMEPR bounds.     

A class of 1-generator quasi-cyclic codes

If R = F/sub q/[x/spl rceil/]/(x/sup m/ - 1), S = F/sub qn/[x]/(x/sup m/ - 1), we define the mapping a_(x) /spl rarr/ A(x) =/spl sigma//sub 0//sup n-1/a/sub i/(x)/spl alpha//sub i/ from R/sup n/ onto S, where (/spl alpha//sub 0/, /spl alpha//sub i/,..., /spl alpha//sub n-1/) is a basis for F/sub qn/ over F/sub q/. This carries the q-ray 1-generator quasicyclic (QC) code R a_(x) onto the code RA(x) in S whose parity-check polynomial (p.c.p.) is defined as the monic polynomial h(x) over F/sub q/ of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of x/sub m/ 1 over F/sub q/ and F/sub qn/ are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal in S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for F/sub qn/ over F/sub q/, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes.