关于等重码不可检错误概率的界

本文研究了二元等重码不可检错误概率(UEP)的界.首先,我们通过研究二元等重码的对偶距离分布及其性质,给出二元等重码UEP的一个新的下界,该下界改进了Fu-Kl  ve-Wei的最新结果;然后,我们指出2003年Fu-Kl  ve-Wei关于二元等重码UEP上界的某些结果有错误,我们随后给出更正后的结果,即二元等重码UEP的平均值和一个上界.     

关于(2k×2k,2k,k×2k)类型非线性等重码的构造

提出利用完美偶阶准二进幻方来直接构造一类典型的(2k×2~k,2~k,k×2~k)非线性等重码问题,分析了完美准二进幻方的结构特点与二进制非线性等重码的映射关系,并具体构造出(16,4,8)二进制非线性等重码。同时指出由2~k×2~k(k≥2)构成的完美偶阶准二进幻方,只要按对称与镜像对称形式划分,均可构成(2k×2~k,2~k,k×2~k)非线性等重码。     

关于二元等重码的最大码字数

本文利用Johnson Schemes理论研究了二元等重码及其最大码字数问题.在Delsarte的associate schemes理论中,Q-变换被引入以研究二元等重码的距离分布.首先,本文研究了等重码距离分布的Q-变换;然后,通过使用Q-变换的性质,我们研究了二元等重码的最大码字数问题并得到码字数的一个新的上界,该上界在形式上类似于纠错码理论中的Grey-Rankin界,并且在某些情况下优于已知的结果.     

关于(2k×2k,2k,k×2k)类型非线性等重码的构造

提出利用完美偶阶准二进幻方来直接构造一类典型的(2k×2k,2k,k×2k)非线性等重码问题,分析了完美准二进幻方的结构特点与二进制非线性等重码的映射关系,并具体构造出(16,4,8)二进制非线性等重码.同时指出由2k×2k(k≥2)构成的完美偶阶准二进幻方,只要按对称与镜像对称形式划分,均可构成(2k×2k,2k,k×2k)非线性等重码.     

一种可用于非线性码译码神经网络模型研究

本文提出一种非线性码神经网络译码方案,在纠错能力范围内对满足码距特性的一般非线性码以零错误概率进行纠错译码,并在检错能力范围内检错。文中具体描述了神经网络模型构造、学习算法及其理论依据。最后通过非线性等重码的译码实例表明此方案的有效性及理论和应用价值。     

广义Hamming重量和等重码

本文将线性码的广义Hamming重量的概念推广到非线性码上去,并导出了一种广义Elias界.对于线性等重码,本文给出了其完整的重量谱系.     

一种可用于非线性码译码神经网络模型研究

本文提出一种非线性码神经网络译码方案。在纠错能力范围内对满足码距特性的一般非线性码以零错误概率进行纠错译码，并在检错能力范围内检错。文中具体描述了神经网络模型构造、学习算法及其理论依据。最后通过非线性等重码的译码实例表明此方案的有铲性及理论和应用价值。     

基于二次乘法特征的射影线性码

基于有限域上的二次乘法特征构造了两类线性码，精确计算出了它们的参数和重量分布.结果表明，第一类线性码是射影三重码，且对偶码关于球填充界几乎最优；第二类线性码是射影二重码，且对偶码关于球填充界几乎最优.此外，本文还得到了一些自正交码和极小码，它们可分别用于构造量子码和安全高效访问结构上的密钥共享方案.     

用于多载波信号峰均功率比抑制的准互补序列集构造方法研究

为解决相互正交互补码集中序列数目受限及多载波码分多址（MC-CDMA）系统信号峰均功率比（PAPR）过高的问题，提出一类具有低列向量PAPR且参数渐进达到最优的非周期二元准互补序列集（QCSS）的构造。通过设计一类新的映射函数集，得到的参数渐进最优的非周期二元QCSS与已有二元QCSS相比具有更多的序列数目。并将正交Golay序列集作为初始矩阵，构造得到的非周期QCSS列向量为Golay序列，进而保证了其列向量PAPR不超过2。实验仿真结果表明，所构造的互补序列集可以有效地将时域MC-CDMA信号PAPR降低到3 dB，同时系统具有良好的误码率性能。     

一类达到上界的拉丁方和正交拉丁方等重码

证明了一类能够达到Johson上界的特殊拉丁方和正交拉丁方所对应的等重码的存在性。讨论了其存在条件,导出了有关基本公式,并给出实例设计和该码的检错性能分析。     

基于FFT-BP算法的多元域LDPC码译码器设计与FPGA实现

GF(q)域上的LDPC码是二进制LDPC码的扩展,它具有比二进制LDPC码更好的纠错性能。FFT-BP算法是高效的LDPC码译码算法,本文在GF(4)域上探讨该算法的设计与实现。本文的创新之处在于,根据FFT-BP算法的特点设计了一种利用Tanner图进行信息索引的方式,简化了地址查询模块的设计。实验表明,在归一化信噪比为2.6dB时,译码器的误码率可达到10-6。     

A characterization of MMD codes

Let C be a linear [n,k,d]-code over GF(q) with k⩾2. If s=n-k+1-d denotes the defect of C, then by the Griesmer bound, d⩽(s+1)q. Now, for obvious reasons, we are interested in codes of given defect s for which the minimum distance is maximal, i.e., d=(s+1)q. We classify up to formal equivalence all such linear codes over GF(q). Remember that two codes over GF(q) are formally equivalent if they have the same weight distribution. It turns out that for k⩾3 such codes exist only in dimension 3 and 4 with the ternary extended Golay code, the ternary dual Golay code, and the binary even-weight code as exceptions. In dimension 4 they are related to ovoids in PG(3,q) except the binary extended Hamming code, and in dimension 3 to maximal arcs in PG(2,q)     

一类新的能够渐进达到Gilbert-Varshamov界的Alternant子类码

本文基于Maximum Distance Separable(MDS)码的Hamming重量分布提出一类新的二元Alternant子类码.分析表明这类新的子类码包含整个BCH码类,并且可以渐进达到Gilbert-Varshamov(GV)界.     

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10^-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.     

Lossy Source Compression of Non‐Uniform Binary Source via Reinforced Belief Propagation over GQ‐LDGM Codes

In this letter, we consider the lossy coding of a non‐uniform binary source based on GF(q)‐quantized low‐density generator matrix (LDGM) codes with check degree d_c=2. By quantizing the GF(q) LDGM codeword, a non‐uniform binary codeword can be obtained, which is suitable for direct quantization of the non‐uniform binary source. Encoding is performed by reinforced belief propagation, a variant of belief propagation. Simulation results show that the performance of our method is quite close to the theoretic rate‐distortion bounds. For example, when the GF(16)‐LDGM code with a rate of 0.4 and block‐length of 1,500 is used to compress the non‐uniform binary source with probability of 1 being 0.23, the distortion is 0.091, which is very close to the optimal theoretical value of 0.074.     

Construction of LDPC codes over GF（q） with modified progressive edge growth

A parity check matrix construction method for constructing a low-density parity-check (LDPC) codes over GF(q) (q>2) based on the modified progressive edge growth (PEG) algorithm is introduced. First, the nonzero locations of the parity check matrix are selected using the PEG algorithm. Then the nonzero elements are defined by avoiding the definition of subcode. A proof is given to show the good minimum distance property of constructed GF(q)-LDPC codes. Simulations are also presented to illustrate the good error performance of the designed codes.     

有限域上两类新的2-重量码的构造

有限域上二重量码的构造是图论、编码与密码中的重要研究课题.本文得到了有限域上两类新的2-重量码并且它们都是最优的,达到了Griesmer界.这些码由有限域的扩域上迹码的p元像定义,有阿贝尔码的代数结构,利用特征和和高斯和来计算了它们的重量分布.我们也计算了这些像码的对偶码的极小距离.最后对扩域上迹码的像在秘钥共享方案中的应用进行了刻画.