关于秩距离BCH码的校验矩阵及其秩距离

本文基于秩距离码提出了秩距离BCH码,给出了其校验矩阵的形式,并讨论了所给秩距离BCH码为最大秩距离HCH码时,码的生成多项式的根应满足的条件。     

基于最大秩距离码的数字签名方案

G.Kabatianskii等在<基于随机纠错码的数字签名方案>一文中,基于随机纠错码构造了一数字签名方案.本文依据上述文章中的构造方法,基于最大秩距离码,构造一新的数字签名方案,说明了该方案比G.Kabatianskii等的方案更实用.     

本原BCH码参数的盲识别方法

针对本原BCH码编码参数的盲识别,首先,根据循环移位前后码字的最大公约式的阶数,利用实际序列与随机序列阶数的概率分布差异最大的特性,提出了一种基于变异系数识别码长的方法,在此基础上,根据码字之间的线性约束关系,以阶数概率最大值为下限,通过计算概率总和来识别起始点,进而,计算邻域半径快速去除含错码字,根据阶数分布最大值识别生成多项式,实现了BCH码的盲识别。理论分析及仿真实验表明,该识别算法简单易行,在误码率为0.01的条件下识别效果较好,容错性较强。     

二进制BCH码的一种盲识别方法

提出二进制BCH码的一种盲识别方法。该算法适用于本原和非本原二进制BCH码。首先,在帧长度已知的条件下,根据循环特性,给出一种分组长度的统计识别方法;然后,根据循环特性及各种约束条件得到备选多项式;再根据校正子权重和最小原则,得到最优多项式;最后通过因式分解得到生成多项式的最终估计表达式。仿真表明,本文算法具有较强的抗随机误码能力,而且其识别性能随着参加统计的码字数增多而提高。该算法不涉及矩阵运算,因此非常适合硬件实现。     

关于Goppa码、BCH码的广义Hamming重量

本文研究了Goppa码、BCH码的广义Hamming重量,给出了Goppa码的广义Hamming重量的一个下界以及求该下界的一个算法;对于本原、狭义BCH码,给出了后面一些广义Hamming重量的确切值。     

关于q元BCH码的维数和最小距离

本文讨论的是ｑ元狭义本原ＢＣＨ码，以下简称ＢＣＧ码。首先给出了一定条件下求ＢＣＧ码维数的一般公式，该结果改进了ＭｗｃＡｉｌｌｉａｍｓ等人（１９７７）的结果。然后给出了求ＢＣＨ码维数的一般迭代方法。     

关于q元BCH码的维数和最小距离

本文讨论的是q元狭义本原BCH码,以下简称BCH码。首先给出了一定条件下求BCH码维数的一般公式,该结果改进了MacWilliams等人(1977)的结果。然后给出了求BCH码维数的一般迭代方法。此外,本文还指出了BCH码的最小距离的BCH界是分圆陪集首,我们猜测BCH码的最小距离也是分圆陪集首。     

关于Goppa码,BCH码的广义Hamming重量

本文研究了Ｇｏｐｐａ码、ＢＣＨ码的广义Ｈａｍｍｉｎｇ重量,给出了Ｇｏｐｐａ码的广义Ｈａｍｍｉｎｇ重量的一个下界以及求该下界的一个算法;对于本原、狭义ＢＣＨ码,给出了后面一些广义Ｈａｍｍｉｎｇ重量的确切值。     

基于最大秩距离码的两种Rao-Nam方案

基于最大秩距离码，提出两种新的Rao-Nam方案，讨论了它们的可行性及安全性，并证明了它们比基于纠错码的Rao-Nam方案更安全。     

On classical BCH codes and quantum BCH codes

It is a regular way of constructing quantum error-correcting codes via codes with self-orthogonal property, and whether a classical Bose-Chaudhuri-Hocquenghem (BCH) code is self-orthogonal can be determined by its designed distance. In this paper, we give the sufficient and necessary condition for arbitrary classical BCH codes with self-orthogonal property through algorithms. We also give a better upper bound of the designed distance of a classical narrow-sense BCH code which contains its Euclidean dual. Besides these, we also give one algorithm to compute the dimension of these codes. The complexity of all algorithms is analyzed. Then the results can be applied to construct a series of quantum BCH codes via the famous CSS constructions.     

卡氏积码的MDR码和自对偶码

定义了Z_(r_1),Z_(r_2)…,Z_(r_s)上线性码C_1,C_2,…,C_s的卡氏积码.利用子模同构定理,研究了在Z_(r_1)×Z_(r_2)×…×Z_(r_s)上卡氏积码C_1×C_2×…×C_s的秩与在Z_(r_1),Z_(r_2),…,Z_(r_s)码C_1,C_2,…,C_s的秩的关系,借助这一关系,得到了MDR码的卡氏积仍为MDR码和自对偶码的卡氏积码也为自对偶码.     

卷积码盲识别方法研究

提出了一种码率删除卷积码的盲识别算法.该算法基于卷积码的线性特性和校验性质,利用一种优化方法求解二元域线性方程组,估计出校验多项式矩阵,并建立删除卷积码的数学变换模型,由校验多项式矩阵估计出删除卷积码的源码生成多项式矩阵和删除模式.     

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.     

LDPC码的最小汉明距离估算

低密度奇偶校验码(LDPC)是一种优秀的线性分组奇偶校验码。在简要阐述LDPC码原理上,给出了一种叫“最小码字搜索法”的算法来估算LDPC码的最小汉明距离。并用相应的实例给予验证,显示了该方法的正确性与实用性,对分析与优化LDPC码设计具有重要参考价值。     

More on the distance distribution of BCH codes

We derive a new estimate for the error term in the binomial approximation to the distance distribution of BCH codes. This is an improvement on the earlier bounds by Kasami-Fujiwara-Lin (1985), Vladuts-Skorobogatov (1991), and Krasikov-Litsyn (1995)     

Gaussian integers and interleaved rank codes for space–time block codes

This paper presents the design of space–time block codes (STBCs) over maximum rank distance (MRD) codes, energy‐efficient STBCs, STBCs using interleaved‐MRD codes, the use of Gaussian integers for STBCs modulation, and Gabidulin's decoding algorithm for decoding STBCs. The design fundamentals of STBCs using MRD codes are firstly put forward for different number of transmit antennas. Extension finite fields (Galois fields) are used to design these linear block codes. Afterward, a comparative study of MRD‐based STBCs with corresponding orthogonal and quasi‐orthogonal codes is also included in the paper. The simulation results show that rank codes, for any number of transmit antennas, exhibit diversity gain at full rate contrary to orthogonal codes, which give diversity gain at full rate only for two transmit antennas case. Secondly, an energy‐efficient MRD‐STBC is proposed, which outperforms orthogonal STBC at least for 2 × 1 antenna system. Thirdly, interleaved‐MRD codes are used to construct higher‐order transmit antenna systems. Using interleaved‐MRD codes further reduces the complexity (compared with normal MRD codes) of the decoding algorithm. Fourthly, the use of Gaussian integers is utilized in mapping MRD‐based STBCs to complex constellations. Furthermore, it is described how an efficient and computationally less complex Gabidulin's decoding algorithm can be exploited for decoding complex MRD‐STBCs. The decoding results have been compared against hard‐decision maximum likelihood decoding. Under this decoding scheme, MRD‐STBCs have been shown to be potential candidate for higher transmit antenna systems as the decoding complexity of Gabidulin's algorithm is far less, and its performance for decoding MRD‐STBCs is somewhat reasonable. Copyright © 2015 John Wiley & Sons, Ltd.