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.     

一种三元线性互补对偶码与自正交码的构造方法

有限域上线性互补对偶(LCD)码有良好的相关特性和正交特性,并能够防御信道攻击。自正交码是编码理论中一类非常重要的码,可以用于构造量子纠错码。该文研究了有限域F₃上的LCD码。通过选取4种合适的定义集,利用有限域F₃上线性码是LCD码或自正交码的判定条件,构造了4类3元LCD码和一些自正交码,并研究了这4类线性码的对偶码,得到了一些3元最优线性码。     

一种四元厄米特LCD码与厄米特自正交码的构造方法

有限域上线性互补对偶（LCD）码具有良好的结构和性质,并在双用户加法器信道中得到了广泛的应用.自正交码是编码理论中一类重要的线性码,常被用于构造量子纠错码.本文根据有限域上线性码是厄米特LCD码或厄米特自正交码的判定条件,通过选取合适的定义集,构造出了四类四元厄米特LCD码和厄米特自正交码.同时,本文还研究了这四类线性码的厄米特对偶码,并得到了一些四元最优线性码.     

有限域上常循环厄密特对偶包含码及其应用

该文研究了有限域GF(q2)上长度为(q2m-1)/(q2-1)的常循环码。给出一类常循环码是厄米特对偶包含码的一个充要条件,并确定了这类常循环厄米特对偶包含码的参数。利用厄米特构造,得到了比量子BCH码参数更好的量子纠错码。     

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.     

F4m上厄米特自正交常循环码

有限域上常循环码具有丰富的代数结构,其编译码电路容易实现,因而在信息传输实践中具有重要的应用.该文研究了一类有限域上任意长度的厄米特自正交常循环码的结构,给出了此类有限域上厄米特自正交常循环码的生成多项式与存在条件,确立了此类有限域上厄米特自正交常循环码的计数公式,并且利用此类有限域上偶长度的厄米特自正交常循环码构造了最优的量子码.     

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.     

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.     

新纠缠辅助量子MDS码的构造

纠缠辅助量子纠错码是经典量子纠错码的推广,通过在接收者和发送者双方预先共享纠缠态的方式实现量子通信.由于预先共享纠缠态会造成额外的费用,如何构造具有较小预先共享纠缠态的纠缠辅助量子纠错码是一个有趣的问题.本文给出了有限域Fq2上一类负循环码是厄米特对偶包含码的充分条件,通过研究其分圆陪集的结构性质,确定了不同数目的预先共享纠缠态的存在条件,并结合纠缠辅助量子纠错码的构造方法,构造了一些新的具有较小预先共享纠缠态的纠缠辅助量子Maximum-Distance-Separable(MDS)码.     

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     

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.     

On lowest density MDS codes

Let F_q denote the finite field GF(q) and let h be a positive integer. MDS (maximum distance separable) codes over the symbol alphabet F_q^b are considered that are linear over F _q and have sparse (“low-density”) parity-check and generator matrices over F_q that are systematic over F_q^b. Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an F_q-linear MDS code over F_q^b, along with upper bounds on the length of any MDS code that attains those lower bounds. A construction is presented that achieves those bounds for certain redundancy values. The building block of the construction is a set of sparse nonsingular matrices over F_q whose pairwise differences are also nonsingular. Bounds and constructions are presented also for the case where the systematic condition on the parity-check and generator matrices is relaxed to be over F_q, rather than over F_q^b     

Chinese product of constacyclic and cyclic codes over finite rings

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.     

Cyclic codes and self-dual codes over F2+uF2

We introduce linear cyclic codes over the ring F₂+uF ₂={0,1,u,u¯=u+1}, where u²=0 and study them by analogy with the Z₄ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z₄-codes. Unlike the Z₄ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map