基于极小线性码上的秘密共享方案

从理论上说,每个线性码都可用于构造秘密共享方案,但是在一般情况下,所构造的秘密共享方案的存取结构是难以确定的.本文提出了极小线性码的概念,指出基于这种码的对偶码所构造的秘密共享方案的存取结构是容易确定的.本文首先证明了极小线性码的缩短码一定是极小线性码.然后对几类不可约循环码给出它们为极小线性码的判定条件,并在理论上研究了基于几类不可约循环码的对偶码上的秘密共享方案的存取结构.最后用编程具体求出了一些实例中方案的存取结构.     

Z4上LCD循环码及其二元像

循环码和线性互补对偶(LCD)码是两类重要的线性码,在数据存储、通信系统和密码等领域有着广泛的应用.本文研究了Z4上奇长度的LCD循环码,给出了Z4上奇长度的循环码为LCD码的一个充要条件,证明了Z4上LCD循环码的二元像是可逆码;构造了Z4上长为2m+1的LCD循环码,得到了参数较好的二元非线性可逆码.     

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。     

一类一步多数逻辑可译码的广义汉明重量谱

线性码的广义汉明重量谱描述了码在第二类窃密信道中传输的密码学特征。该文针对一类循环码在仿射置换群之下不变的一步多数逻辑可译码的广义汉明重量谱进行了研究,提出了该类码的重量谱的估计方法,并通过实例作了说明。     

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。     

两类极小二元线性码的构造

线性码在数据存储、信息安全以及秘密共享等领域具有重要的作用。而极小线性码是设计秘密共享方案的首选码,设计极小线性码是当前密码与编码研究的重要内容之一。该文首先选取恰当的布尔函数,研究了函数的Walsh谱值分布,并利用布尔函数的Walsh谱值分布构造了两类极小线性码,确定了码的参数及重量分布。结果表明,所构造的码是不满足Ashikhmin-Barg条件的极小线性码,可用作设计具有良好访问结构的秘密共享方案。     

有限非链环上的自对偶常循环码及其应用

纠错码是提高信息传输效率与可靠性的重要手段.构造性能良好的线性码类是纠错码研究中的一个基本问题.本文主要讨论了有限非链环F_q[v]/（v^m－v）上自对偶常循环码的代数结构,包括Euclidean自对偶常循环码、Hermitian自对偶常循环码以及Hermitian自对偶常循环码的极大距离可分（MDS）码.本文给出了环F_q[v]/（v^m－v）上常循环码是Euclidean自对偶码的充分条件,以及是Hermitian自对偶码的充要条件,并利用Gray映射构造了有限域F_q上一些参数较好的自对偶码.特别地,本文得到了有限域F_19²上一个新的参数为[16,8,6]的Hermitian自对偶码.     

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。     

一类p元最优线性码和低相关性线性序列的构造

在信息理论中,最优线性码具有很强的纠错能力、低相关性线性序列在密码系统和CDMA通信系统中得到了广泛应用.因此构造最优线性码和构造低相关性线性序列具有重要的研究价值.记R=Fp+uFp,这里的p为奇素数.本文首先通过迹映射构造出环R上的一类新的线性码,然后将这类新的线性码的删余码通过Gray映射得到了域Fp上一类最优码.同时,通过迹映射构造出环R上的一类线性循环码,将这类线性循环码视为线性周期序列并通过广义Nechaev-Gray映射得到了域Fp上一类低相关线性周期序列.     

环Z_4上线性循环码的深度谱

Etzion定义并研究了域Fq上线性码的深度谱,该文研究了环Z4上线性码与线性循环码的深度谱,证明了4k12k2型线性码的深度谱至少含有k1 k2个非零值,并给出了一类4k型线性循环码的深度谱为{n,n-1,…,n-k 1}。     

Transactions Papers - Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach

This paper is concerned with construction of efficiently encodable nonbinary quasi-cyclic LDPC codes based on finite fields. Four classes of nonbinary quasi-cyclic LDPC codes are constructed. Experimental results show that codes constructed perform well with iterative decoding using a fast Fourier transform based q-ary sum-product algorithm and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard- decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm.     

A Unified Approach to the Construction of Binary and Nonbinary Quasi-Cyclic LDPC Codes Based on Finite Fields

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.     

基于一些PBIBD的低密度校验码的构造

本文构造了两类部分平衡不完全区组设计.并利用它们构造了一类低密度校验码(LDPC码),其最小环长至少为6,码率的选取具有很大的灵活性,而且可以具有拟循环结构.计算机仿真结果表明这种方法构造的LDPC码,在加性高斯白噪声信道中BPSK调制下用和积迭代译码性能很好.     

基于素域构造的准循环低密度校验码

该文提出一种基于素域构造准循环低密度校验码的方法。该方法是Lan等所提出基于有限域构造准循环低密度校验码的方法在素域上的推广,给出了一类更广泛的基于素域构造的准循环低密度校验码。通过仿真结果证实:所构造的这一类准循环低密度校验码在高斯白噪声信道上采用迭代译码时具有优良的纠错性能。     

Low-density parity-check codes based on finite geometries: arediscovery and new results

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding     

Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.     

High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2. The construction is similar to the derivation of Reed-Solomon (RS)-based LDPC codes presented by Chen and Djurdjevic Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes     

Cyclotomic Linear Codes of Order $3$

In this correspondence, two classes of cyclotomic linear codes over GF(q) of order 3 are constructed and their weight distributions are determined. The two classes are two-weight codes and contain optimal codes. They are not equivalent to irreducible cyclic codes in general when q > 2.     

Construction of nonbinary cyclic, quasi-cyclic and regular LDPC codes: a finite geometry approach

This paper presents five methods for constructing nonbinary LDPC codes based on finite geometries. These methods result in five classes of nonbinary LDPC codes, one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. Experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the AWGN channel and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates with either algebraic hard-decision decoding or Kotter-Vardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity.